Groups, Rings And Fields

# Groups, Rings And Fields

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A1.4 B1.3

comment(i) Let $R$ be a commutative ring. Define the terms prime ideal and maximal ideal, and show that if an ideal $M$ in $R$ is maximal then $M$ is also prime.

(ii) Let $P$ be a non-trivial prime ideal in the commutative ring $R$ ('non-trivial' meaning that $P \neq\{0\}$ and $P \neq R$ ). If $P$ has finite index as a subgroup of $R$, show that $P$ is also maximal. Give an example to show that this may fail, if the assumption of finite index is omitted. Finally, show that if $R$ is a principal ideal domain, then every non-trivial prime ideal in $R$ is maximal.

A2.4 B2.3

comment(i) State Gauss' Lemma on polynomial irreducibility. State and prove Eisenstein's criterion.

(ii) Which of the following polynomials are irreducible over $\mathbb{Q}$ ? Justify your answers.

(a) $x^{7}-3 x^{3}+18 x+12$

(b) $x^{4}-4 x^{3}+11 x^{2}-3 x-5$

(c) $1+x+x^{2}+\ldots+x^{p-1}$ with $p$ prime

[Hint: consider substituting $y=x-1$.]

(d) $x^{n}+p x+p^{2}$ with $p$ prime.

[Hint: show any factor has degree at least two, and consider powers of $p$ dividing coefficients.]

A3.4

comment(i) Let $K \leqslant \mathbb{C}$ be a field and $L \leqslant \mathbb{C}$ a finite normal extension of $K$. If $H$ is a finite subgroup of order $m$ in the Galois group $G(L \mid K)$, show that $L$ is a normal extension of the $H$-invariant subfield $I(H)$ of degree $m$ and that $G(L \mid I(H))=H$. [You may assume the theorem of the primitive element.]

(ii) Show that the splitting field over $\mathbb{Q}$ of the polynomial $x^{4}+2$ is $\mathbb{Q}[\sqrt[4]{2}, i]$ and deduce that its Galois group has order 8. Exhibit a subgroup of order 4 of the Galois group, and determine the corresponding invariant subfield.

A4.4

comment(a) Let $t$ be the maximal power of the prime $p$ dividing the order of the finite group $G$, and let $N\left(p^{t}\right)$ denote the number of subgroups of $G$ of order $p^{t}$. State clearly the numerical restrictions on $N\left(p^{t}\right)$ given by the Sylow theorems.

If $H$ and $K$ are subgroups of $G$ of orders $r$ and $s$ respectively, and their intersection $H \cap K$ has order $t$, show the set $H K=\{h k: h \in H, k \in K\}$ contains $r s / t$ elements.

(b) The finite group $G$ has 48 elements. By computing the possible values of $N(16)$, show that $G$ cannot be simple.

A1.4

comment(i) Let $p$ be a prime number. Show that a group $G$ of order $p^{n}(n \geqslant 2)$ has a nontrivial normal subgroup, that is, $G$ is not a simple group.

(ii) Let $p$ and $q$ be primes, $p>q$. Show that a group $G$ of order $p q$ has a normal Sylow $p$-subgroup. If $G$ has also a normal Sylow $q$-subgroup, show that $G$ is cyclic. Give a necessary and sufficient condition on $p$ and $q$ for the existence of a non-abelian group of order $p q$. Justify your answer.

A2.4 B2.3

comment(i) In each of the following two cases, determine a highest common factor in $\mathbb{Z}[i]$ :

(a) $3+4 i, 4-3 i$;

(b) $3+4 i, 1+2 i$.

(ii) State and prove the Eisenstein criterion for irreducibility of polynomials with integer coefficients. Show that, if $p$ is prime, the polynomial

$1+x+\cdots+x^{p-1}$

is irreducible over $\mathbb{Z}$.

A3.4

comment(i) Let $K$ be the splitting field of the polynomial $f=X^{3}-2$ over the rationals. Find the Galois group $G$ of $K / \mathbb{Q}$ and describe its action on the roots of $f$.

(ii) Let $K$ be the splitting field of the polynomial $X^{4}+a X^{2}+b$ (where $a, b \in \mathbb{Q}$ ) over the rationals. Assuming that the polynomial is irreducible, prove that the Galois group $G$ of the extension $K / \mathbb{Q}$ is either $C_{4}$, or $C_{2} \times C_{2}$, or the dihedral group $D_{8}$.

A4.4

commentWrite an essay on the theory of invariants. Your essay should discuss the theorem on the finite generation of the ring of invariants, the theorem on elementary symmetric functions, and some examples of calculation of rings of invariants.

B1.3

comment(i) Let $p$ be a prime number. Show that a group $G$ of order $p^{n}(n \geqslant 2)$ has a nontrivial normal subgroup, that is, $G$ is not a simple group.

(ii) Let $p$ and $q$ be primes, $p>q$. Show that a group $G$ of order $p q$ has a normal Sylow $p$-subgroup. If $G$ has also a normal Sylow $q$-subgroup, show that $G$ is cyclic. Give a necessary and sufficient condition on $p$ and $q$ for the existence of a non-abelian group of order $p q$. Justify your answer.

A1.4

comment(i) What is a Sylow subgroup? State Sylow's Theorems.

Show that any group of order 33 is cyclic.

(ii) Prove the existence part of Sylow's Theorems.

[You may use without proof any arithmetic results about binomial coefficients which you need.]

Show that a group of order $p^{2} q$, where $p$ and $q$ are distinct primes, is not simple. Is it always abelian? Give a proof or a counterexample.

A2.4 B2.3

comment(i) Show that the ring $\mathbb{Z}[i]$ is Euclidean.

(ii) What are the units in $\mathbb{Z}[i]$ ? What are the primes in $\mathbb{Z}[i]$ ? Justify your answers. Factorize $11+7 i$ into primes in $\mathbb{Z}[i]$.

A3.4

comment(i) What does it mean for a ring to be Noetherian? State Hilbert's Basis Theorem. Give an example of a Noetherian ring which is not a principal ideal domain.

(ii) Prove Hilbert's Basis Theorem.

Is it true that if the ring $R[X]$ is Noetherian, then so is $R ?$

A4.4

commentLet $F$ be a finite field. Show that there is a unique prime $p$ for which $F$ contains the field $\mathbb{F}_{p}$ of $p$ elements. Prove that $F$ contains $p^{n}$ elements, for some $n \in \mathbb{N}$. Show that $x^{p^{n}}=x$ for all $x \in F$, and hence find a polynomial $f \in \mathbb{F}_{p}[X]$ such that $F$ is the splitting field of $f$. Show that, up to isomorphism, $F$ is the unique field $\mathbb{F}_{p^{n}}$ of size $p^{n}$.

[Standard results about splitting fields may be assumed.]

Prove that the mapping sending $x$ to $x^{p}$ is an automorphism of $\mathbb{F}_{p^{n}}$. Deduce that the Galois group Gal $\left(\mathbb{F}_{p^{n}} / \mathbb{F}_{p}\right)$ is cyclic of order $n$. For which $m$ is $\mathbb{F}_{p^{m}}$ a subfield of $\mathbb{F}_{p^{n}}$ ?

B1.3

commentState Sylow's Theorems. Prove the existence part of Sylow's Theorems.

Show that any group of order 33 is cyclic.

Show that a group of order $p^{2} q$, where $p$ and $q$ are distinct primes, is not simple. Is it always abelian? Give a proof or a counterexample.

A1.4 B1.3

comment(i) Define the notion of a Sylow $p$-subgroup of a finite group $G$, and state a theorem concerning the number of them and the relation between them.

(ii) Show that any group of order 30 has a non-trivial normal subgroup. Is it true that every group of order 30 is commutative?

A2.4 B2.3

comment(i) Show that the ring $k=\mathbf{F}_{2}[X] /\left(X^{2}+X+1\right)$ is a field. How many elements does it have?

(ii) Let $k$ be as in (i). By considering what happens to a chosen basis of the vector space $k^{2}$, or otherwise, find the order of the groups $G L_{2}(k)$ and $S L_{2}(k)$.

By considering the set of lines in $k^{2}$, or otherwise, show that $S L_{2}(k)$ is a subgroup of the symmetric group $S_{5}$, and identify this subgroup.

A3.4

comment(i) Let $G$ be the cyclic subgroup of $G L_{2}(\mathbf{C})$ generated by the matrix $\left(\begin{array}{cc}1 & 2 \\ 0 & -1\end{array}\right)$, acting on the polynomial ring $\mathbf{C}[X, Y]$. Determine the ring of invariants $\mathbf{C}[X, Y]^{G}$.

(ii) Determine $\mathbf{C}[X, Y]^{G}$ when $G$ is the cyclic group generated by $\left(\begin{array}{cc}0 & -1 \\ 1 & -1\end{array}\right)$.

[Hint: consider the eigenvectors.]

A4.4

commentShow that the ring $\mathbf{Z}[\omega]$ is Euclidean, where $\omega=\exp (2 \pi i / 3)$.

Show that a prime number $p \neq 3$ is reducible in $\mathbf{Z}[\omega]$ if and only if $p \equiv 1(\bmod 3)$.

Which prime numbers $p$ can be written in the form $p=a^{2}+a b+b^{2}$ with $a, b \in \mathbf{Z}$ (and why)?