# Galois Theory

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Paper 1, Section II, 18I

comment(a) Let $K \subseteq L$ be fields, and $f(x) \in K[x]$ a polynomial.

Define what it means for $L$ to be a splitting field for $f$ over $K$.

Prove that splitting fields exist, and state precisely the theorem on uniqueness of splitting fields.

Let $f(x)=x^{3}-2 \in \mathbb{Q}[x]$. Find a subfield of $\mathbb{C}$ which is a splitting field for $f$ over Q. Is this subfield unique? Justify your answer.

(b) Let $L=\mathbb{Q}\left[\zeta_{7}\right]$, where $\zeta_{7}$ is a primitive 7 th root of unity.

Show that the extension $L / \mathbb{Q}$ is Galois. Determine all subfields $M \subseteq L$.

For each subfield $M$, find a primitive element for the extension $M / \mathbb{Q}$ explicitly in terms of $\zeta_{7}$, find its minimal polynomial, and write $\operatorname{down} \operatorname{Aut}(M / \mathbb{Q})$ and $\operatorname{Aut}(L / M)$.

Which of these subfields $M$ are Galois over $\mathbb{Q}$ ?

[You may assume the Galois correspondence, but should prove any results you need about cyclotomic extensions directly.]

Paper 2, Section II, 18I

comment(a) Let $f(x) \in \mathbb{F}_{q}[x]$ be a polynomial of degree $n$, and let $L$ be its splitting field.

(i) Suppose that $f$ is irreducible. Compute $\operatorname{Gal}(f)$, carefully stating any theorems you use.

(ii) Now suppose that $f(x)$ factors as $f=h_{1} \cdots h_{r}$ in $\mathbb{F}_{q}[x]$, with each $h_{i}$ irreducible, and $h_{i} \neq h_{j}$ if $i \neq j$. Compute $\operatorname{Gal}(f)$, carefully stating any theorems you use.

(iii) Explain why $L / \mathbb{F}_{q}$ is a cyclotomic extension. Define the corresponding homomorphism $\operatorname{Gal}\left(L / \mathbb{F}_{q}\right) \hookrightarrow(\mathbb{Z} / m \mathbb{Z})^{*}$ for this extension (for a suitable integer $m$ ), and compute its image.

(b) Compute $\operatorname{Gal}(f)$ for the polynomial $f=x^{4}+8 x+12 \in \mathbb{Q}[x]$. [You may assume that $f$ is irreducible and that its discriminant is $576^{2}$.]

Paper 3, Section II, 18I

commentDefine the elementary symmetric functions in the variables $x_{1}, \ldots, x_{n}$. State the fundamental theorem of symmetric functions.

Let $f(x)=x^{n}+a_{n-1} x^{n-1}+\cdots+a_{0} \in K[x]$, where $K$ is a field. Define the discriminant of $f$, and explain why it is a polynomial in $a_{0}, \ldots, a_{n-1}$.

Compute the discriminant of $x^{5}+q$.

Let $f(x)=x^{5}+p x^{2}+q$. When does the discriminant of $f(x)$ equal zero? Compute the discriminant of $f(x)$.

Paper 4 , Section II, 18I

commentLet $L$ be a field, and $G$ a group which acts on $L$ by field automorphisms.

(a) Explain the meaning of the phrase in italics in the previous sentence.

Show that the set $L^{G}$ of fixed points is a subfield of $L$.

(b) Suppose that $G$ is finite, and set $K=L^{G}$. Let $\alpha \in L$. Show that $\alpha$ is algebraic and separable over $K$, and that the degree of $\alpha$ over $K$ divides the order of $G$.

Assume that $\alpha$ is a primitive element for the extension $L / K$, and that $G$ is a subgroup of $\operatorname{Aut}(L)$. What is the degree of $\alpha$ over $K$ ? Justify your answer.

(c) Let $L=\mathbb{C}(z)$, and let $\zeta_{n}$ be a primitive $n$th root of unity in $\mathbb{C}$ for some integer $n>1$. Show that the $\mathbb{C}$-automorphisms $\sigma, \tau$ of $L$ defined by

$\sigma(z)=\zeta_{n} z, \quad \tau(z)=1 / z$

generate a group $G$ isomorphic to the dihedral group of order $2 n$.

Find an element $w \in L$ for which $L^{G}=\mathbb{C}(w)$.

Paper 1, Section II, 18G

comment(a) State and prove the tower law.

(b) Let $K$ be a field and let $f(x) \in K[x]$.

(i) Define what it means for an extension $L / K$ to be a splitting field for $f$.

(ii) Suppose $f$ is irreducible in $K[x]$, and char $K=0$. Let $M / K$ be an extension of fields. Show that the roots of $f$ in $M$ are distinct.

(iii) Let $h(x)=x^{q^{n}}-x \in K[x]$, where $K=F_{q}$ is the finite field with $q$ elements. Let $L$ be a splitting field for $h$. Show that the roots of $h$ in $L$ are distinct. Show that $[L: K]=n$. Show that if $f(x) \in K[x]$ is irreducible, and deg $f=n$, then $f$ divides $x^{q^{n}}-x$.

(iv) For each prime $p$, give an example of a field $K$, and a polynomial $f(x) \in K[x]$ of degree $p$, so that $f$ has at most one root in any extension $L$ of $K$, with multiplicity $p$.

Paper 2, Section II, 18G

comment(a) Let $K$ be a field and let $L$ be the splitting field of a polynomial $f(x) \in K[x]$. Let $\xi_{N}$ be a primitive $N^{\text {th }}$root of unity. Show that $\operatorname{Aut}\left(L\left(\xi_{N}\right) / K\left(\xi_{N}\right)\right)$ is a subgroup of $\operatorname{Aut}(L / K)$.

(b) Suppose that $L / K$ is a Galois extension of fields with cyclic Galois group generated by an element $\sigma$ of order $d$, and that $K$ contains a primitive $d^{\text {th }}$root of unity $\xi_{d}$. Show that an eigenvector $\alpha$ for $\sigma$ on $L$ with eigenvalue $\xi_{d}$ generates $L / K$, that is, $L=K(\alpha)$. Show that $\alpha^{d} \in K$.

(c) Let $G$ be a finite group. Define what it means for $G$ to be solvable.

Determine whether

(i) $G=S_{4} ; \quad$ (ii) $G=S_{5}$

are solvable.

(d) Let $K=\mathbb{Q}\left(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}\right)$ be the field of fractions of the polynomial ring $\mathbb{Q}\left[a_{1}, a_{2}, a_{3}, a_{4}, a_{5}\right]$. Let $f(x)=x^{5}-a_{1} x^{4}+a_{2} x^{3}-a_{3} x^{2}+a_{4} x-a_{5} \in K[x]$. Show that $f$ is not solvable by radicals. [You may use results from the course provided that you state them clearly.]

Paper 3, Section II, 18G

comment(a) Let $L / K$ be a Galois extension of fields, with $\operatorname{Aut}(L / K)=A_{10}$, the alternating group on 10 elements. Find $[L: K]$.

Let $f(x)=x^{2}+b x+c \in K[x]$ be an irreducible polynomial, char $K \neq 2$. Show that $f(x)$ remains irreducible in $L[x] .$

(b) Let $L=\mathbb{Q}\left[\xi_{11}\right]$, where $\xi_{11}$ is a primitive $11^{\text {th }}$root of unity.

Determine all subfields $M \subseteq L$. Which are Galois over $\mathbb{Q}$ ?

For each proper subfield $M$, show that an element in $M$ which is not in $\mathbb{Q}$ must be primitive, and give an example of such an element explicitly in terms of $\xi_{11}$ for each $M$. [You do not need to justify that your examples are not in $\mathbb{Q}$.]

Find a primitive element for the extension $L / \mathbb{Q}$.

Paper 4, Section II, 18G

comment(a) Let $K$ be a field. Define the discriminant $\Delta(f)$ of a polynomial $f(x) \in K[x]$, and explain why it is in $K$, carefully stating any theorems you use.

Compute the discriminant of $x^{4}+r x+s$.

(b) Let $K$ be a field and let $f(x) \in K[x]$ be a quartic polynomial with roots $\alpha_{1}, \ldots, \alpha_{4}$ such that $\alpha_{1}+\cdots+\alpha_{4}=0$.

Define the resolvant cubic $g(x)$ of $f(x)$.

Suppose that $\Delta(f)$ is a square in $K$. Prove that the resolvant cubic is irreducible if and only if $G a l(f)=A_{4}$. Determine the possible Galois groups Gal $(f)$ if $g(x)$ is reducible.

The resolvant cubic of $x^{4}+r x+s$ is $x^{3}-4 s x-r^{2}$. Using this, or otherwise, determine $\operatorname{Gal}(f)$, where $f(x)=x^{4}+8 x+12 \in \mathbb{Q}[x]$. [You may use without proof that $f$ is irreducible.]

Paper 1, Section II, 18F

comment(a) Suppose $K, L$ are fields and $\sigma_{1}, \ldots, \sigma_{m}$ are distinct embeddings of $K$ into $L$. Prove that there do not exist elements $\lambda_{1}, \ldots, \lambda_{m}$ of $L$ (not all zero) such that

$\lambda_{1} \sigma_{1}(x)+\cdots+\lambda_{m} \sigma_{m}(x)=0 \quad \text { for all } x \in K$

(b) For a finite field extension $K$ of a field $k$ and for $\sigma_{1}, \ldots, \sigma_{m}$ distinct $k$ automorphisms of $K$, show that $m \leqslant[K: k]$. In particular, if $G$ is a finite group of field automorphisms of a field $K$ with $K^{G}$ the fixed field, deduce that $|G| \leqslant\left[K: K^{G}\right]$.

(c) If $K=\mathbb{Q}(x, y)$ with $x, y$ independent transcendentals over $\mathbb{Q}$, consider the group $G$ generated by automorphisms $\sigma$ and $\tau$ of $K$, where

$\sigma(x)=y, \sigma(y)=-x \quad \text { and } \quad \tau(x)=x, \tau(y)=-y .$

Prove that $|G|=8$ and that $K^{G}=\mathbb{Q}\left(x^{2}+y^{2}, x^{2} y^{2}\right)$.

Paper 2, Section II, F

commentFor any prime $p \neq 5$, explain briefly why the Galois group of $X^{5}-1$ over $\mathbb{F}_{p}$ is cyclic of order $d$, where $d=1$ if $p \equiv 1 \bmod 5, d=4$ if $p \equiv 2,3 \bmod 5$, and $d=2$ if $p \equiv 4$ $\bmod 5 .$

Show that the splitting field of $X^{5}-5$ over $\mathbb{Q}$ is an extension of degree 20 .

For any prime $p \neq 5$, prove that $X^{5}-5 \in \mathbb{F}_{p}[X]$ does not have an irreducible cubic as a factor. For $p \equiv 2$ or $3 \bmod 5$, show that $X^{5}-5$ is the product of a linear factor and an irreducible quartic over $\mathbb{F}_{p}$. For $p \equiv 1 \bmod 5$, show that either $X^{5}-5$ is irreducible over $\mathbb{F}_{p}$ or it splits completely.

[You may assume the reduction mod p criterion for finding cycle types in the Galois group of a monic polynomial over $\mathbb{Z}$ and standard facts about finite fields.]

Paper 3, Section II, F

commentLet $k$ be a field. For $m$ a positive integer, consider $X^{m}-1 \in k[X]$, where either char $k=0$, or char $k=p$ with $p$ not dividing $m$; explain why the polynomial has distinct roots in a splitting field.

For $m$ a positive integer, define the $m$ th cyclotomic polynomial $\Phi_{m} \in \mathbb{C}[X]$ and show that it is a monic polynomial in $\mathbb{Z}[X]$. Prove that $\Phi_{m}$ is irreducible over $\mathbb{Q}$ for all $m$. [Hint: If $\Phi_{m}=f g$, with $f, g \in \mathbb{Z}[X]$ and $f$ monic irreducible with $0<\operatorname{deg} f<\operatorname{deg} \Phi_{m}$, and $\varepsilon$ is a root of $f$, show first that $\varepsilon^{p}$ is a root of $f$ for any prime $p$ not dividing $m$.]

Let $F=X^{8}+X^{7}-X^{5}-X^{4}-X^{3}+X+1 \in \mathbb{Z}[X]$; by considering the product $\left(X^{2}-X+1\right) F$, or otherwise, show that $F$ is irreducible over $\mathbb{Q}$.

Paper 4, Section II, $18 \mathrm{~F}$

commentState (without proof) a result concerning uniqueness of splitting fields of a polynomial.

Given a polynomial $f \in \mathbb{Q}[X]$ with distinct roots, what is meant by its Galois group $\operatorname{Gal}_{\mathbb{Q}}(f)$ ? Show that $f$ is irreducible over $\mathbb{Q}$ if and only if $\mathrm{Gal}_{\mathbb{Q}}(f)$ acts transitively on the roots of $f$.

Now consider an irreducible quartic of the form $g(X)=X^{4}+b X^{2}+c \in \mathbb{Q}[X]$. If $\alpha \in \mathbb{C}$ denotes a root of $g$, show that the splitting field $K \subset \mathbb{C}$ is $\mathbb{Q}(\alpha, \sqrt{c})$. Give an explicit description of $\operatorname{Gal}(K / \mathbb{Q})$ in the cases:

(i) $\sqrt{c} \in \mathbb{Q}(\alpha)$, and

(ii) $\sqrt{c} \notin \mathbb{Q}(\alpha)$.

If $c$ is a square in $\mathbb{Q}$, deduce that $\operatorname{Gal}_{\mathbb{Q}}(g) \cong C_{2} \times C_{2}$. Conversely, if Gal $_{\mathbb{Q}}(g) \cong$ $C_{2} \times C_{2}$, show that $\sqrt{c}$ is invariant under at least two elements of order two in the Galois group, and deduce that $c$ is a square in $\mathbb{Q}$.

Paper 1, Section II, I

commentLet $f(t)=t^{4}+b t^{2}+c t+d$ be an irreducible quartic with rational coefficients. Explain briefly why it is that if the cubic $g(t)=t^{3}+2 b t^{2}+\left(b^{2}-4 d\right) t-c^{2}$ has $S_{3}$ as its Galois group then the Galois group of $f(t)$ is $S_{4}$.

For which prime numbers $p$ is the Galois group of $t^{4}+p t+p$ a proper subgroup of $S_{4}$ ? [You may assume that the discriminant of $t^{3}+\lambda t+\mu$ is $-4 \lambda^{3}-27 \mu^{2}$.]

Paper 2, Section II, I

commentLet $K$ be a field and let $f(t)$ be a monic polynomial with coefficients in $K$. What is meant by a splitting field $L$ for $f(t)$ over $K$ ? Show that such a splitting field exists and is unique up to isomorphism.

Now suppose that $K$ is a finite field. Prove that $L$ is a Galois extension of $K$ with cyclic Galois group. Prove also that the degree of $L$ over $K$ is equal to the least common multiple of the degrees of the irreducible factors of $f(t)$ over $K$.

Now suppose $K$ is the field with two elements, and let

$\mathcal{P}_{n}=\{f(t) \in K[t] \mid f \text { has degree } n \text { and is irreducible over } K\}$

How many elements does the set $\mathcal{P}_{9}$ have?

Paper 3, Section II, I

commentLet $L$ be a finite field extension of a field $K$, and let $G$ be a finite group of $K$ automorphisms of $L$. Denote by $L^{G}$ the field of elements of $L$ fixed by the action of $G$.

(a) Prove that the degree of $L$ over $L^{G}$ is equal to the order of the group $G$.

(b) For any $\alpha \in L$ write $f(t, \alpha)=\Pi_{g \in G}(t-g(\alpha))$.

(i) Suppose that $L=K(\alpha)$. Prove that the coefficients of $f(t, \alpha)$ generate $L^{G}$ over $K$.

(ii) Suppose that $L=K\left(\alpha_{1}, \alpha_{2}\right)$. Prove that the coefficients of $f\left(t, \alpha_{1}\right)$ and $f\left(t, \alpha_{2}\right)$ lie in $L^{G}$. By considering the case $L=K\left(a_{1}^{1 / 2}, a_{2}^{1 / 2}\right)$ with $a_{1}$ and $a_{2}$ in $K$, or otherwise, show that they need not generate $L^{G}$ over $K$.

Paper 4, Section II, I

commentLet $K$ be a field of characteristic $p>0$ and let $L$ be the splitting field of the polynomial $f(t)=t^{p}-t+a$ over $K$, where $a \in K$. Let $\alpha \in L$ be a root of $f(t)$.

If $L \neq K$, show that $f(t)$ is irreducible over $K$, that $L=K(\alpha)$, and that $L$ is a Galois extension of $K$. What is $\operatorname{Gal}(L / K)$ ?

Paper 1, Section II, I

comment(a) Let $K$ be a field and let $f(t) \in K[t]$. What does it mean for a field extension $L$ of $K$ to be a splitting field for $f(t)$ over $K$ ?

Show that the splitting field for $f(t)$ over $K$ is unique up to isomorphism.

(b) Find the Galois groups over the rationals $\mathbb{Q}$ for the following polynomials: (i) $t^{4}+2 t+2$. (ii) $t^{5}-t-1$.

Paper 2, Section II, I

comment(a) Define what it means for a finite field extension $L$ of a field $K$ to be separable. Show that $L$ is of the form $K(\alpha)$ for some $\alpha \in L$.

(b) Let $p$ and $q$ be distinct prime numbers. Let $L=\mathbb{Q}(\sqrt{p}, \sqrt{-q})$. Express $L$ in the form $\mathbb{Q}(\alpha)$ and find the minimal polynomial of $\alpha$ over $\mathbb{Q}$.

(c) Give an example of a field extension $K \leqslant L$ of finite degree, where $L$ is not of the form $K(\alpha)$. Justify your answer.

Paper 3, Section II, I

comment(a) Let $F$ be a finite field of characteristic $p$. Show that $F$ is a finite Galois extension of the field $F_{p}$ of $p$ elements, and that the Galois group of $F$ over $F_{p}$ is cyclic.

(b) Find the Galois groups of the following polynomials:

(i) $t^{4}+1$ over $F_{3}$.

(ii) $t^{3}-t-2$ over $F_{5}$.

(iii) $t^{4}-1$ over $F_{7}$.

Paper 4, Section II, I

comment(a) State the Fundamental Theorem of Galois Theory.

(b) What does it mean for an extension $L$ of $\mathbb{Q}$ to be cyclotomic? Show that a cyclotomic extension $L$ of $\mathbb{Q}$ is a Galois extension and prove that its Galois group is Abelian.

(c) What is the Galois group $G$ of $\mathbb{Q}(\eta)$ over $\mathbb{Q}$, where $\eta$ is a primitive 7 th root of unity? Identify the intermediate subfields $M$, with $\mathbb{Q} \leqslant M \leqslant \mathbb{Q}(\eta)$, in terms of $\eta$, and identify subgroups of $G$ to which they correspond. Justify your answers.

Paper 1, Section II, H

comment(a) Prove that if $K$ is a field and $f \in K[t]$, then there exists a splitting field $L$ of $f$ over $K$. [You do not need to show uniqueness of $L$.]

(b) Let $K_{1}$ and $K_{2}$ be algebraically closed fields of the same characteristic. Show that either $K_{1}$ is isomorphic to a subfield of $K_{2}$ or $K_{2}$ is isomorphic to a subfield of $K_{1}$. [For subfields $F_{i}$ of $K_{1}$ and field homomorphisms $\psi_{i}: F_{i} \rightarrow K_{2}$ with $i=1$, 2, we say $\left(F_{1}, \psi_{1}\right) \leqslant\left(F_{2}, \psi_{2}\right)$ if $F_{1}$ is a subfield of $F_{2}$ and $\left.\psi_{2}\right|_{F_{1}}=\psi_{1}$. You may assume the existence of a maximal pair $(F, \psi)$ with respect to the partial order just defined.]

(c) Give an example of a finite field extension $K \subseteq L$ such that there exist $\alpha, \beta \in L \backslash K$ where $\alpha$ is separable over $K$ but $\beta$ is not separable over $K .$

Paper 2, Section II, H

comment(a) Let $K \subseteq L$ be a finite separable field extension. Show that there exist only finitely many intermediate fields $K \subseteq F \subseteq L$.

(b) Define what is meant by a normal extension. Is $\mathbb{Q} \subseteq \mathbb{Q}(\sqrt{1+\sqrt{7}})$ a normal extension? Justify your answer.

(c) Prove Artin's lemma, which states: if $K \subseteq L$ is a field extension, $H$ is a finite subgroup of $\operatorname{Aut}_{K}(L)$, and $F:=L^{H}$ is the fixed field of $H$, then $F \subseteq L$ is a Galois extension with $\operatorname{Gal}(L / F)=H$.

Paper 3, Section II, H

comment(a) Let $L$ be the 13 th cyclotomic extension of $\mathbb{Q}$, and let $\mu$ be a 13 th primitive root of unity. What is the minimal polynomial of $\mu$ over $\mathbb{Q}$ ? What is the Galois group $\operatorname{Gal}(L / \mathbb{Q})$ ? Put $\lambda=\mu+\frac{1}{\mu}$. Show that $\mathbb{Q} \subseteq \mathbb{Q}(\lambda)$ is a Galois extension and find $\operatorname{Gal}(\mathbb{Q}(\lambda) / \mathbb{Q})$.

(b) Define what is meant by a Kummer extension. Let $K$ be a field of characteristic zero and let $L$ be the $n$th cyclotomic extension of $K$. Show that there is a sequence of Kummer extensions $K=F_{1} \subseteq F_{2} \subseteq \cdots \subseteq F_{r}$ such that $L$ is contained in $F_{r}$.

Paper 4, Section II, H

comment(a) Let $f=t^{5}-9 t+3 \in \mathbb{Q}[t]$ and let $L$ be the splitting field of $f$ over $\mathbb{Q}$. Show that $\operatorname{Gal}(L / \mathbb{Q})$ is isomorphic to $S_{5}$. Let $\alpha$ be a root of $f$. Show that $\mathbb{Q} \subseteq \mathbb{Q}(\alpha)$ is neither a radical extension nor a solvable extension.

(b) Let $f=t^{26}+2$ and let $L$ be the splitting field of $f$ over $\mathbb{Q}$. Is it true that $\operatorname{Gal}(L / \mathbb{Q})$ has an element of order 29 ? Justify your answer. Using reduction mod $p$ techniques, or otherwise, show that $\operatorname{Gal}(L / \mathbb{Q})$ has an element of order 3 .

[Standard results from the course may be used provided they are clearly stated.]

Paper 1, Section II, $17 \mathrm{~F}$

comment(i) Let $K \subseteq L$ be a field extension and $f \in K[t]$ be irreducible of positive degree. Prove the theorem which states that there is a $1-1$ correspondence

$\operatorname{Root}_{f}(L) \longleftrightarrow \operatorname{Hom}_{K}\left(\frac{K[t]}{\langle f\rangle}, L\right)$

(ii) Let $K$ be a field and $f \in K[t]$. What is a splitting field for $f$ ? What does it mean to say $f$ is separable? Show that every $f \in K[t]$ is separable if $K$ is a finite field.

(iii) The primitive element theorem states that if $K \subseteq L$ is a finite separable field extension, then $L=K(\alpha)$ for some $\alpha \in L$. Give the proof of this theorem assuming $K$ is infinite.

Paper 2, Section II, F

comment(i) State the fundamental theorem of Galois theory, without proof. Let $L$ be a splitting field of $t^{3}-2 \in \mathbb{Q}[t]$. Show that $\mathbb{Q} \subseteq L$ is Galois and that Gal $(L / \mathbb{Q})$ has a subgroup which is not normal.

(ii) Let $\Phi_{8}$ be the 8 th cyclotomic polynomial and denote its image in $\mathbb{F}_{7}[t]$ again by $\Phi_{8}$. Show that $\Phi_{8}$ is not irreducible in $\mathbb{F}_{7}[t]$.

(iii) Let $m$ and $n$ be coprime natural numbers, and let $\mu_{m}=\exp (2 \pi i / m)$ and $\mu_{n}=\exp (2 \pi i / n)$ where $i=\sqrt{-1}$. Show that $\mathbb{Q}\left(\mu_{m}\right) \cap \mathbb{Q}\left(\mu_{n}\right)=\mathbb{Q}$.

Paper 3, Section II, F

commentLet $f \in \mathbb{Q}[t]$ be of degree $n>0$, with no repeated roots, and let $L$ be a splitting field for $f$.

(i) Show that $f$ is irreducible if and only if for any $\alpha, \beta \in \operatorname{Root}_{f}(L)$ there is $\phi \in \operatorname{Gal}(L / \mathbb{Q})$ such that $\phi(\alpha)=\beta$.

(ii) Explain how to define an injective homomorphism $\tau: \operatorname{Gal}(L / \mathbb{Q}) \rightarrow S_{n}$. Find an example in which the image of $\tau$ is the subgroup of $S_{3}$ generated by (2 3). Find another example in which $\tau$ is an isomorphism onto $S_{3}$.

(iii) Let $f(t)=t^{5}-3$ and assume $f$ is irreducible. Find a chain of subgroups of $\operatorname{Gal}(L / \mathbb{Q})$ that shows it is a solvable group. [You may quote without proof any theorems from the course, provided you state them clearly.]

Paper 4, Section II, $17 \mathrm{~F}$

comment(i) Prove that a finite solvable extension $K \subseteq L$ of fields of characteristic zero is a radical extension.

(ii) Let $x_{1}, \ldots, x_{7}$ be variables, $L=\mathbb{Q}\left(x_{1}, \ldots, x_{7}\right)$, and $K=\mathbb{Q}\left(e_{1}, \ldots, e_{7}\right)$ where $e_{i}$ are the elementary symmetric polynomials in the variables $x_{i}$. Is there an element $\alpha \in L$ such that $\alpha^{2} \in K$ but $\alpha \notin K$ ? Justify your answer.

(iii) Find an example of a field extension $K \subseteq L$ of degree two such that $L \neq K(\sqrt{\alpha})$ for any $\alpha \in K$. Give an example of a field which has no extension containing an $11 t h$ primitive root of unity.

Paper 1, Section II, 18H

commentWhat is meant by the statement that $L$ is a splitting field for $f \in K[X] ?$

Show that if $f \in K[X]$, then there exists a splitting field for $f$ over $K$. Explain the sense in which a splitting field for $f$ over $K$ is unique.

Determine the degree $[L: K]$ of a splitting field $L$ of the polynomial $f=X^{4}-4 X^{2}+2$ over $K$ in the cases (i) $K=\mathbb{Q}$, (ii) $K=\mathbb{F}_{5}$, and (iii) $K=\mathbb{F}_{7}$.

Paper 2, Section II, H

commentDescribe the Galois correspondence for a finite Galois extension $L / K$.

Let $L$ be the splitting field of $X^{4}-2$ over $\mathbb{Q}$. Compute the Galois group $G$ of $L / \mathbb{Q}$. For each subgroup of $G$, determine the corresponding subfield of $L$.

Let $L / K$ be a finite Galois extension whose Galois group is isomorphic to $S_{n}$. Show that $L$ is the splitting field of a separable polynomial of degree $n$.

Paper 3, Section II, H

commentLet $L / K$ be an algebraic extension of fields, and $x \in L$. What does it mean to say that $x$ is separable over $K$ ? What does it mean to say that $L / K$ is separable?

Let $K=\mathbb{F}_{p}(t)$ be the field of rational functions over $\mathbb{F}_{p}$.

(i) Show that if $x$ is inseparable over $K$ then $K(x)$ contains a $p$ th root of $t$.

(ii) Show that if $L / K$ is finite there exists $n \geqslant 0$ and $y \in L$ such that $y^{p^{n}}=t$ and $L / K(y)$ is separable.

Show that $Y^{2}+t Y+t$ is an irreducible separable polynomial over the field of rational functions $K=\mathbb{F}_{2}(t)$. Find the degree of the splitting field of $X^{4}+t X^{2}+t$ over $K$.

Paper 4, Section II, H

comment(i) Let $G$ be a finite subgroup of the multiplicative group of a field. Show that $G$ is cyclic.

(ii) Let $\Phi_{n}(X)$ be the $n$th cyclotomic polynomial. Let $p$ be a prime not dividing $n$, and let $L$ be a splitting field for $\Phi_{n}$ over $\mathbb{F}_{p}$. Show that $L$ has $p^{m}$ elements, where $m$ is the least positive integer such that $p^{m} \equiv 1(\bmod n)$.

(iii) Find the degrees of the irreducible factors of $X^{35}-1$ over $\mathbb{F}_{2}$, and the number of factors of each degree.

Paper 1, Section II, I

comment(i) Give an example of a field $F$, contained in $\mathbb{C}$, such that $X^{4}+1$ is a product of two irreducible quadratic polynomials in $F[X]$. Justify your answer.

(ii) Let $F$ be any extension of degree 3 over $\mathbb{Q}$. Prove that the polynomial $X^{4}+1$ is irreducible over $F$.

(iii) Give an example of a prime number $p$ such that $X^{4}+1$ is a product of two irreducible quadratic polynomials in $\mathbb{F}_{p}[X]$. Justify your answer.

[Standard facts on fields, extensions, and finite fields may be quoted without proof, as long as they are stated clearly.]

Paper 2, Section II, I

commentFor a positive integer $N$, let $\mathbb{Q}\left(\boldsymbol{\mu}_{N}\right)$ be the cyclotomic field obtained by adjoining all $N$-th roots of unity to $\mathbb{Q}$. Let $F=\mathbb{Q}\left(\boldsymbol{\mu}_{24}\right)$.

(i) Determine the Galois group of $F$ over $\mathbb{Q}$.

(ii) Find all $N>1$ such that $\mathbb{Q}\left(\boldsymbol{\mu}_{N}\right)$ is contained in $F$.

(iii) List all quadratic and quartic extensions of $\mathbb{Q}$ which are contained in $F$, in the form $\mathbb{Q}(\alpha)$ or $\mathbb{Q}(\alpha, \beta)$. Indicate which of these fields occurred in (ii).

[Standard facts on the Galois groups of cyclotomic fields and the fundamental theorem of Galois theory may be used freely without proof.]

Paper 3, Section II, I

commentLet $p$ be a prime number and $F$ a field of characteristic $p$. Let $\operatorname{Fr}_{p}: F \rightarrow F$ be the Frobenius map defined by $\operatorname{Fr}_{p}(x)=x^{p}$ for all $x \in F$.

(i) Prove that $\operatorname{Fr}_{p}$ is a field automorphism when $F$ is a finite field.

(ii) Is the same true for an arbitrary algebraic extension $F$ of $\mathbb{F}_{p}$ ? Justify your answer.

(iii) Let $F=\mathbb{F}_{p}\left(X_{1}, \ldots, X_{n}\right)$ be the rational function field in $n$ variables where $n \geqslant 1$ over $\mathbb{F}_{p}$. Determine the image of $\operatorname{Fr}_{p}: F \rightarrow F$, and show that $\operatorname{Fr}_{p}$ makes $F$ into an extension of degree $p^{n}$ over a subfield isomorphic to $F$. Is it a separable extension?

Paper 4, Section II, I

comment(i) Let $\zeta_{N}=\mathrm{e}^{2 \pi i / N} \in \mathbb{C}$ for $N \geqslant 1$. For the cases $N=11,13$, is it possible to express $\zeta_{N}$, starting with integers and using rational functions and (possibly nested) radicals? If it is possible, briefly explain how this is done, assuming standard facts in Galois Theory.

(ii) Let $F=\mathbb{C}(X, Y, Z)$ be the rational function field in three variables over $\mathbb{C}$, and for integers $a, b, c \geqslant 1$ let $K=\mathbb{C}\left(X^{a}, Y^{b}, Z^{c}\right)$ be the subfield of $F$ consisting of all rational functions in $X^{a}, Y^{b}, Z^{c}$ with coefficients in $\mathbb{C}$. Show that $F / K$ is Galois, and determine its Galois group. [Hint: For $\alpha, \beta, \gamma \in \mathbb{C}^{\times}$, the map $(X, Y, Z) \longmapsto(\alpha X, \beta Y, \gamma Z)$ is an automorphism of $F$.]

Paper 1, Section II, 18H

commentList all subfields of the cyclotomic field $\mathbb{Q}\left(\boldsymbol{\mu}_{20}\right)$ obtained by adjoining all 20 th roots of unity to $\mathbb{Q}$, and draw the lattice diagram of inclusions among them. Write all the subfields in the form $\mathbb{Q}(\alpha)$ or $\mathbb{Q}(\alpha, \beta)$. Briefly justify your answer.

[The description of the Galois group of cyclotomic fields and the fundamental theorem of Galois theory can be used freely without proof.]

Paper 2, Section II, H

commentLet $K, L$ be subfields of $\mathbb{C}$ with $K \subset L$.

Suppose that $K$ is contained in $\mathbb{R}$ and $L / K$ is a finite Galois extension of odd degree. Prove that $L$ is also contained in $\mathbb{R}$.

Give one concrete example of $K, L$ as above with $K \neq L$. Also give an example in which $K$ is contained in $\mathbb{R}$ and $L / K$ has odd degree, but is not Galois and $L$ is not contained in $\mathbb{R}$.

[Standard facts on fields and their extensions can be quoted without proof, as long as they are clearly stated.]

Paper 3, Section II, H

commentLet $q=p^{f}(f \geqslant 1)$ be a power of the prime $p$, and $\mathbb{F}_{q}$ be a finite field consisting of $q$ elements.

Let $N$ be a positive integer prime to $p$, and $\mathbb{F}_{q}\left(\boldsymbol{\mu}_{N}\right)$ be the cyclotomic extension obtained by adjoining all $N$ th roots of unity to $\mathbb{F}_{q}$. Prove that $\mathbb{F}_{q}\left(\boldsymbol{\mu}_{N}\right)$ is a finite field with $q^{n}$ elements, where $n$ is the order of the element $q \bmod N$ in the multiplicative group $(\mathbb{Z} / N \mathbb{Z})^{\times}$of the $\operatorname{ring} \mathbb{Z} / N \mathbb{Z}$.

Explain why what is proven above specialises to the following fact: the finite field $\mathbb{F}_{p}$ for an odd prime $p$ contains a square root of $-1$ if and only if $p \equiv 1(\bmod 4)$.

[Standard facts on finite fields and their extensions can be quoted without proof, as long as they are clearly stated.]

Paper 4, Section II, H

commentLet $F=\mathbb{C}\left(X_{1}, \ldots, X_{n}\right)$ be a field of rational functions in $n$ variables over $\mathbb{C}$, and let $s_{1}, \ldots, s_{n}$ be the elementary symmetric polynomials:

$s_{j}:=\sum_{\left\{i_{1}, \ldots, i_{j}\right\} \subset\{1, \ldots, n\}} X_{i_{1}} \cdots X_{i_{j}} \in F \quad(1 \leqslant j \leqslant n),$

and let $K=\mathbb{C}\left(s_{1}, \ldots, s_{n}\right)$ be the subfield of $F$ generated by $s_{1}, \ldots, s_{n}$. Let $1 \leqslant m \leqslant n$, and $Y:=X_{1}+\cdots+X_{m} \in F$. Let $K(Y)$ be the subfield of $F$ generated by $Y$ over $K$. Find the degree $[K(Y): K]$.

[Standard facts about the fields $F, K$ and Galois extensions can be quoted without proof, as long as they are clearly stated.]

Paper 1, Section II, 18H

commentLet $K$ be a field.

(i) Let $F$ and $F^{\prime}$ be two finite extensions of $K$. When the degrees of these two extensions are equal, show that every $K$-homomorphism $F \rightarrow F^{\prime}$ is an isomorphism. Give an example, with justification, of two finite extensions $F$ and $F^{\prime}$ of $K$, which have the same degrees but are not isomorphic over $K$.

(ii) Let $L$ be a finite extension of $K$. Let $F$ and $F^{\prime}$ be two finite extensions of $L$. Show that if $F$ and $F^{\prime}$ are isomorphic as extensions of $L$ then they are isomorphic as extensions of $K$. Prove or disprove the converse.

Paper 2, Section II, H

commentLet $F=\mathbb{C}(x, y)$ be the function field in two variables $x, y$. Let $n \geqslant 1$, and $K=\mathbb{C}\left(x^{n}+y^{n}, x y\right)$ be the subfield of $F$ of all rational functions in $x^{n}+y^{n}$ and $x y .$

(i) Let $K^{\prime}=K\left(x^{n}\right)$, which is a subfield of $F$. Show that $K^{\prime} / K$ is a quadratic extension.

(ii) Show that $F / K^{\prime}$ is cyclic of order $n$, and $F / K$ is Galois. Determine the Galois $\operatorname{group} \operatorname{Gal}(F / K)$.

Paper 3, Section II, H

commentLet $n \geqslant 1$ and $K=\mathbb{Q}\left(\boldsymbol{\mu}_{n}\right)$ be the cyclotomic field generated by the $n$th roots of unity. Let $a \in \mathbb{Q}$ with $a \neq 0$, and consider $F=K(\sqrt[n]{a})$.

(i) State, without proof, the theorem which determines $\operatorname{Gal}(K / \mathbb{Q})$.

(ii) Show that $F / \mathbb{Q}$ is a Galois extension and that $\operatorname{Gal}(F / \mathbb{Q})$ is soluble. [When using facts about general Galois extensions and their generators, you should state them clearly.]

(iii) When $n=p$ is prime, list all possible degrees $[F: \mathbb{Q}]$, with justification.

Paper 4, Section II, H

commentLet $K$ be a field of characteristic 0 , and let $P(X)=X^{4}+b X^{2}+c X+d$ be an irreducible quartic polynomial over $K$. Let $\alpha_{1}, \alpha_{2}, \alpha_{3}, \alpha_{4}$ be its roots in an algebraic closure of $K$, and consider the Galois group $\operatorname{Gal}(P)$ (the group $\operatorname{Gal}(F / K)$ for a splitting field $F$ of $P$ over $K$ ) as a subgroup of $S_{4}$ (the group of permutations of $\left.\alpha_{1}, \alpha_{2}, \alpha_{3}, \alpha_{4}\right)$.

Suppose that $\operatorname{Gal}(P)$ contains $V_{4}=\{1,(12)(34),(13)(24),(14)(23)\}$.

(i) List all possible $\operatorname{Gal}(P)$ up to isomorphism. [Hint: there are 4 cases, with orders 4 , 8,12 and 24.]

(ii) Let $Q(X)$ be the resolvent cubic of $P$, i.e. a cubic in $K[X]$ whose roots are $-\left(\alpha_{1}+\alpha_{2}\right)\left(\alpha_{3}+\alpha_{4}\right),-\left(\alpha_{1}+\alpha_{3}\right)\left(\alpha_{2}+\alpha_{4}\right)$ and $-\left(\alpha_{1}+\alpha_{4}\right)\left(\alpha_{2}+\alpha_{3}\right)$. Construct a natural surjection $\operatorname{Gal}(P) \rightarrow \operatorname{Gal}(Q)$, and find $\operatorname{Gal}(Q)$ in each of the four cases found in (i).

(iii) Let $\Delta \in K$ be the discriminant of $Q$. Give a criterion to determine $\operatorname{Gal}(P)$ in terms of $\Delta$ and the factorisation of $Q$ in $K[X]$.

(iv) Give a specific example of $P$ where $\operatorname{Gal}(P)$ is abelian.

Paper 1, Section II, 18H

commentLet $\mathbb{F}_{q}$ be a finite field with $q$ elements and $\overline{\mathbb{F}}_{q}$ its algebraic closure.

(i) Give a non-zero polynomial $P(X)$ in $\mathbb{F}_{q}\left[X_{1}, \ldots, X_{n}\right]$ such that

$P\left(\alpha_{1}, \ldots, \alpha_{n}\right)=0 \quad \text { for all } \alpha_{1}, \ldots, \alpha_{n} \in \mathbb{F}_{q}$

(ii) Show that every irreducible polynomial $P(X)$ of degree $n>0$ in $\mathbb{F}_{q}[X]$ can be factored in $\overline{\mathbb{F}}_{q}[X]$ as $(X-\alpha)\left(X-\alpha^{q}\right)\left(X-\alpha^{q^{2}}\right) \cdots\left(X-\alpha^{q^{n-1}}\right)$ for some $\alpha \in \overline{\mathbb{F}}_{q}$. What is the splitting field and the Galois group of $P$ over $\mathbb{F}_{q}$ ?

(iii) Let $n$ be a positive integer and $\Phi_{n}(X)$ be the $n$-th cyclotomic polynomial. Recall that if $K$ is a field of characteristic prime to $n$, then the set of all roots of $\Phi_{n}$ in $K$ is precisely the set of all primitive $n$-th roots of unity in $K$. Using this fact, prove that if $p$ is a prime number not dividing $n$, then $p$ divides $\Phi_{n}(x)$ in $\mathbb{Z}$ for some $x \in \mathbb{Z}$ if and only if $p=a n+1$ for some integer $a$. Write down $\Phi_{n}$ explicitly for three different values of $n$ larger than 2 , and give an example of $x$ and $p$ as above for each $n$.

Paper 2, Section II, H

comment(1) Let $F=\mathbb{Q}(\sqrt[3]{5}, \sqrt{5}, i)$. What is the degree of $F / \mathbb{Q}$ ? Justify your answer.

(2) Let $F$ be a splitting field of $X^{4}-5$ over $\mathbb{Q}$. Determine the Galois group $\operatorname{Gal}(F / \mathbb{Q})$. Determine all the subextensions of $F / \mathbb{Q}$, expressing each in the form $\mathbb{Q}(x)$ or $\mathbb{Q}(x, y)$ for some $x, y \in F$.

[Hint: If an automorphism $\rho$ of a field $X$ has order 2 , then for every $x \in X$ the element $x+\rho(x)$ is fixed by $\rho$.]

Paper 3, Section II, H

commentLet $K$ be a field of characteristic 0 . It is known that soluble extensions of $K$ are contained in a succession of cyclotomic and Kummer extensions. We will refine this statement.

Let $n$ be a positive integer. The $n$-th cyclotomic field over a field $K$ is denoted by $K\left(\boldsymbol{\mu}_{n}\right)$. Let $\zeta_{n}$ be a primitive $n$-th root of unity in $K\left(\boldsymbol{\mu}_{n}\right)$.

(i) Write $\zeta_{3} \in \mathbb{Q}\left(\boldsymbol{\mu}_{3}\right), \zeta_{5} \in \mathbb{Q}\left(\boldsymbol{\mu}_{5}\right)$ in terms of radicals. Write $\mathbb{Q}\left(\boldsymbol{\mu}_{3}\right) / \mathbb{Q}$ and $\mathbb{Q}\left(\boldsymbol{\mu}_{5}\right) / \mathbb{Q}$ as a succession of Kummer extensions.

(ii) Let $n>1$, and $F:=K\left(\zeta_{1}, \zeta_{2}, \ldots, \zeta_{n-1}\right)$. Show that $F\left(\boldsymbol{\mu}_{n}\right) / F$ can be written as a succession of Kummer extensions, using the structure theorem of finite abelian groups (in other words, roots of unity can be written in terms of radicals). Show that every soluble extension of $K$ is contained in a succession of Kummer extensions.

Paper 4, Section II, H

commentLet $K$ be a field of characteristic $\neq 2,3$, and assume that $K$ contains a primitive cubic root of unity $\zeta$. Let $P \in K[X]$ be an irreducible cubic polynomial, and let $\alpha, \beta, \gamma$ be its roots in the splitting field $F$ of $P$ over $K$. Recall that the Lagrange resolvent $x$ of $P$ is defined as $x=\alpha+\zeta \beta+\zeta^{2} \gamma$.

(i) List the possibilities for the group $\operatorname{Gal}(F / K)$, and write out the set $\{\sigma(x) \mid \sigma \in \operatorname{Gal}(F / K)\}$ in each case.

(ii) Let $y=\alpha+\zeta \gamma+\zeta^{2} \beta$. Explain why $x^{3}, y^{3}$ must be roots of a quadratic polynomial in $K[X]$. Compute this polynomial for $P=X^{3}+b X+c$, and deduce the criterion to identify $\operatorname{Gal}(F / K)$ through the element $-4 b^{3}-27 c^{2}$ of $K$.

Paper 1, Section II, H

commentDefine a $K$-isomorphism, $\varphi: L \rightarrow L^{\prime}$, where $L, L^{\prime}$ are fields containing a field $K$, and define $\operatorname{Aut}_{K}(L)$.

Suppose $\alpha$ and $\beta$ are algebraic over $K$. Show that $K(\alpha)$ and $K(\beta)$ are $K$-isomorphic via an isomorphism mapping $\alpha$ to $\beta$ if and only if $\alpha$ and $\beta$ have the same minimal polynomial.

Show that $\operatorname{Aut}_{K} K(\alpha)$ is finite, and a subgroup of the symmetric group $S_{d}$, where $d$ is the degree of $\alpha$.

Give an example of a field $K$ of characteristic $p>0$ and $\alpha$ and $\beta$ of the same degree, such that $K(\alpha)$ is not isomorphic to $K(\beta)$. Does such an example exist if $K$ is finite? Justify your answer.

Paper 2, Section II, H

commentFor each of the following polynomials over $\mathbb{Q}$, determine the splitting field $K$ and the Galois group $G$. (1) $x^{4}-2 x^{2}-25$. (2) $x^{4}-2 x^{2}+25$.

Paper 3, Section II, H

commentLet $K=\mathbb{F}_{p}(x)$, the function field in one variable, and let $G=\mathbb{F}_{p}$. The group $G$ acts as automorphisms of $K$ by $\sigma_{a}(x)=x+a$. Show that $K^{G}=\mathbb{F}_{p}(y)$, where $y=x^{p}-x$.

[State clearly any theorems you use.]

Is $K / K^{G}$ a separable extension?

Now let

$H=\left\{\left(\begin{array}{ll} d & a \\ 0 & 1 \end{array}\right): a \in \mathbb{F}_{p}, d \in \mathbb{F}_{p}^{*}\right\}$

and let $H$ act on $K$ by $\left(\begin{array}{ll}d & a \\ 0 & 1\end{array}\right) x=d x+a$. (The group structure on $H$ is given by matrix multiplication.) Compute $K^{H}$. Describe your answer in the form $\mathbb{F}_{p}(z)$ for an explicit $z \in K$.

Is $K^{G} / K^{H}$ a Galois extension? Find the minimum polynomial for $y$ over the field $K^{H}$.

Paper 4, Section II, H

comment(a) Let $K$ be a field. State what it means for $\xi_{n} \in K$ to be a primitive $n$th root of unity.

Show that if $\xi_{n}$ is a primitive $n$th root of unity, then the characteristic of $K$ does not divide $n$. Prove any theorems you use.

(b) Determine the minimum polynomial of a primitive 10 th root of unity $\xi_{10}$ over $\mathbb{Q}$.

Show that $\sqrt{5} \in \mathbb{Q}\left(\xi_{10}\right)$.

(c) Determine $\mathbb{F}_{3}\left(\xi_{10}\right), \mathbb{F}_{11}\left(\xi_{10}\right), \mathbb{F}_{19}\left(\xi_{10}\right)$.

[Hint: Write a necessary and sufficient condition on $q$ for a finite field $\mathbb{F}_{q}$ to contain a primitive 10 th root of unity.]

1.II.18H

commentFind the Galois group of the polynomial $f(x)=x^{4}+x^{3}+1$ over (i) the finite field $\mathbf{F}_{2}$, (ii) the finite field $\mathbf{F}_{3}$, (iii) the finite field $\mathbf{F}_{4}$, (iv) the field $\mathbf{Q}$ of rational numbers.

[Results from the course which you use should be stated precisely.]

2.II.18H

comment(i) Let $K$ be a field, $\theta \in K$, and $n>0$ not divisible by the characteristic. Suppose that $K$ contains a primitive $n$th root of unity. Show that the splitting field of $x^{n}-\theta$ has cyclic Galois group.

(ii) Let $L / K$ be a Galois extension of fields and $\zeta_{n}$ denote a primitive $n$th root of unity in some extension of $L$, where $n$ is not divisible by the characteristic. Show that $\operatorname{Aut}\left(L\left(\zeta_{n}\right) / K\left(\zeta_{n}\right)\right)$ is a subgroup of $\operatorname{Aut}(L / K)$.

(iii) Determine the minimal polynomial of a primitive 6 th root of unity $\zeta_{6}$ over $\mathbf{Q}$.

Compute the Galois group of $x^{6}+3 \in \mathbf{Q}[x]$.

3.II.18H

commentLet $L / K$ be a field extension.

(a) State what it means for $\alpha \in L$ to be algebraic over $K$, and define its degree $\operatorname{deg}_{K}(\alpha)$. Show that if $\operatorname{deg}_{K}(\alpha)$ is odd, then $K(\alpha)=K\left(\alpha^{2}\right)$.

[You may assume any standard results.]

Show directly from the definitions that if $\alpha, \beta \in L$ are algebraic over $K$, then so too is $\alpha+\beta$.

(b) State what it means for $\alpha \in L$ to be separable over $K$, and for the extension $L / K$ to be separable.

Give an example of an inseparable extension $L / K$.

Show that an extension $L / K$ is separable if $L$ is a finite field.

4.II.18H

commentLet $L=\mathbf{C}(z)$ be the function field in one variable, $n>0$ an integer, and $\zeta_{n}=e^{2 \pi i / n}$.

Define $\sigma, \tau: L \rightarrow L$ by the formulae

$(\sigma f)(z)=f\left(\zeta_{n} z\right), \quad(\tau f)(z)=f(1 / z),$

and let $G=\langle\sigma, \tau\rangle$ be the group generated by $\sigma$ and $\tau$.

(i) Find $w \in \mathbf{C}(z)$ such that $L^{G}=\mathbf{C}(w)$.

[You must justify your answer, stating clearly any theorems you use.]

(ii) Suppose $n$ is an odd prime. Determine the subgroups of $G$ and the corresponding intermediate subfields $M$, with $\mathbf{C}(w) \subseteq M \subseteq L$.

State which intermediate subfields $M$ are Galois extensions of $\mathbf{C}(w)$, and for these extensions determine the Galois group.

1.II.18F

commentLet $L / K / M$ be field extensions. Define the degree $[K: M]$ of the field extension $K / M$, and state and prove the tower law.

Now let $K$ be a finite field. Show $\# K=p^{n}$, for some prime $p$ and positive integer $n$. Show also that $K$ contains a subfield of order $p^{m}$ if and only if $m \mid n$.

If $f \in K[x]$ is an irreducible polynomial of degree $d$ over the finite field $K$, determine its Galois group.

2.II.18F

Let $L=K\left(\xi_{n}\right)$, where $\xi_{n}$ is a primitive $n$th root of unity and $G=\operatorname{Aut}(L / K)$. Prove that there is an injective group homomorphism $\chi: G \rightarrow(\mathbb{Z} / n \mathbb{Z})^{*}$.

Show that, if $M$ is an intermediate subfield of $K\left(\xi_{n}\right) / K$, then $M / K$ is Galois. State carefully any results that you use.

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