# Number Fields

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Paper 1, Section II, 20G

commentLet $K=\mathbb{Q}(\alpha)$, where $\alpha^{3}=5 \alpha-8$

(a) Show that $[K: \mathbb{Q}]=3$.

(b) Let $\beta=\left(\alpha+\alpha^{2}\right) / 2$. By considering the matrix of $\beta$ acting on $K$ by multiplication, or otherwise, show that $\beta$ is an algebraic integer, and that $(1, \alpha, \beta)$ is a $\mathbb{Z}$-basis for $\mathcal{O}_{K} \cdot$ [The discriminant of $T^{3}-5 T+8$ is $-4 \cdot 307$, and 307 is prime.]

(c) Compute the prime factorisation of the ideal (3) in $\mathcal{O}_{K}$. Is (2) a prime ideal of $\mathcal{O}_{K} ?$ Justify your answer.

Paper 2, Section II, 20G

commentLet $K$ be a field containing $\mathbb{Q}$. What does it mean to say that an element of $K$ is algebraic? Show that if $\alpha \in K$ is algebraic and non-zero, then there exists $\beta \in \mathbb{Z}[\alpha]$ such that $\alpha \beta$ is a non-zero (rational) integer.

Now let $K$ be a number field, with ring of integers $\mathcal{O}_{K}$. Let $R$ be a subring of $\mathcal{O}_{K}$ whose field of fractions equals $K$. Show that every element of $K$ can be written as $r / m$, where $r \in R$ and $m$ is a positive integer.

Prove that $R$ is a free abelian group of $\operatorname{rank}[K: \mathbb{Q}]$, and that $R$ has finite index in $\mathcal{O}_{K}$. Show also that for every nonzero ideal $I$ of $R$, the index $(R: I)$ of $I$ in $R$ is finite, and that for some positive integer $m, m \mathcal{O}_{K}$ is an ideal of $R$.

Suppose that for every pair of non-zero ideals $I, J \subset R$, we have

$(R: I J)=(R: I)(R: J) .$

Show that $R=\mathcal{O}_{K}$.

[You may assume without proof that $\mathcal{O}_{K}$ is a free abelian group of rank $[K: \mathbb{Q}]$ ] ]

Paper 4 , Section II, 20G

comment(a) Compute the class group of $K=\mathbb{Q}(\sqrt{30})$. Find also the fundamental unit of $K$, stating clearly any general results you use.

[The Minkowski bound for a real quadratic field is $\left|d_{K}\right|^{1 / 2} / 2 .$ ]

(b) Let $K=\mathbb{Q}(\sqrt{d})$ be real quadratic, with embeddings $\sigma_{1}, \sigma_{2} \hookrightarrow \mathbb{R}$. An element $\alpha \in K$ is totally positive if $\sigma_{1}(\alpha)>0$ and $\sigma_{2}(\alpha)>0$. Show that the totally positive elements of $K$ form a subgroup of the multiplicative group $K^{*}$ of index 4 .

Let $I, J \subset \mathcal{O}_{K}$ be non-zero ideals. We say that $I$ is narrowly equivalent to $J$ if there exists a totally positive element $\alpha$ of $K$ such that $I=\alpha J$. Show that this is an equivalence relation, and that the equivalence classes form a group under multiplication. Show also that the order of this group equals

$\begin{cases}\text { the class number } h_{K} \text { of } K & \text { if the fundamental unit of } K \text { has norm }-1 \\ 2 h_{K} & \text { otherwise. }\end{cases}$

Paper 1, Section II, 20G

commentState Minkowski's theorem.

Let $K=\mathbb{Q}(\sqrt{-d})$, where $d$ is a square-free positive integer, not congruent to 3 $(\bmod 4) .$ Show that every nonzero ideal $I \subset \mathcal{O}_{K}$ contains an element $\alpha$ with

$0<\left|N_{K / \mathbb{Q}}(\alpha)\right| \leqslant \frac{4 \sqrt{d}}{\pi} N(I)$

Deduce the finiteness of the class group of $K$.

Compute the class group of $\mathbb{Q}(\sqrt{-22})$. Hence show that the equation $y^{3}=x^{2}+22$ has no integer solutions.

Paper 2, Section II, 20G

comment(a) Let $K$ be a number field of degree $n$. Define the discriminant $\operatorname{disc}\left(\alpha_{1}, \ldots, \alpha_{n}\right)$ of an $n$-tuple of elements $\alpha_{i}$ of $K$, and show that it is nonzero if and only if $\alpha_{1}, \ldots, \alpha_{n}$ is a $\mathbb{Q}$-basis for $K$.

(b) Let $K=\mathbb{Q}(\alpha)$ where $\alpha$ has minimal polynomial

$T^{n}+\sum_{j=0}^{n-1} a_{j} T^{j}, \quad a_{j} \in \mathbb{Z}$

and assume that $p$ is a prime such that, for every $j, a_{j} \equiv 0(\bmod p)$, but $a_{0} \not \equiv 0\left(\bmod p^{2}\right)$.

(i) Show that $P=(p, \alpha)$ is a prime ideal, that $P^{n}=(p)$ and that $\alpha \notin P^{2}$. [Do not assume that $\mathcal{O}_{K}=\mathbb{Z}[\alpha]$.]

(ii) Show that the index of $\mathbb{Z}[\alpha]$ in $\mathcal{O}_{K}$ is prime to $p$.

(iii) If $K=\mathbb{Q}(\alpha)$ with $\alpha^{3}+3 \alpha+3=0$, show that $\mathcal{O}_{K}=\mathbb{Z}[\alpha]$. [You may assume without proof that the discriminant of $T^{3}+a T+b$ is $-4 a^{3}-27 b^{2}$.]

Paper 4, Section II, 20G

commentLet $K$ be a number field of degree $n$, and let $\left\{\sigma_{i}: K \hookrightarrow \mathbb{C}\right\}$ be the set of complex embeddings of $K$. Show that if $\alpha \in \mathcal{O}_{K}$ satisfies $\left|\sigma_{i}(\alpha)\right|=1$ for every $i$, then $\alpha$ is a root of unity. Prove also that there exists $c>1$ such that if $\alpha \in \mathcal{O}_{K}$ and $\left|\sigma_{i}(\alpha)\right|<c$ for all $i$, then $\alpha$ is a root of unity.

State Dirichlet's Unit theorem.

Let $K \subset \mathbb{R}$ be a real quadratic field. Assuming Dirichlet's Unit theorem, show that the set of units of $K$ which are greater than 1 has a smallest element $\epsilon$, and that the group of units of $K$ is then $\left\{\pm \epsilon^{n} \mid n \in \mathbb{Z}\right\}$. Determine $\epsilon$ for $\mathbb{Q}(\sqrt{11})$, justifying your result. [If you use the continued fraction algorithm, you must prove it in full.]

Paper 1, Section II, 20G

commentLet $K=\mathbb{Q}(\sqrt{2})$.

(a) Write down the ring of integers $\mathcal{O}_{K}$.

(b) State Dirichlet's unit theorem, and use it to determine all elements of the group of units $\mathcal{O}_{K}^{\times}$.

(c) Let $P \subset \mathcal{O}_{K}$ denote the ideal generated by $3+\sqrt{2}$. Show that the group

$G=\left\{\alpha \in \mathcal{O}_{K}^{\times} \mid \alpha \equiv 1 \bmod P\right\}$

is cyclic, and find a generator.

Paper 2, Section II, G

comment(a) Let $L$ be a number field. State Minkowski's upper bound for the norm of a representative for a given class of the ideal class group $\mathrm{Cl}\left(\mathcal{O}_{L}\right)$.

(b) Now let $K=\mathbb{Q}(\sqrt{-47})$ and $\omega=\frac{1}{2}(1+\sqrt{-47})$. Using Dedekind's criterion, or otherwise, factorise the ideals $(\omega)$ and $(2+\omega)$ as products of non-zero prime ideals of $\mathcal{O}_{K}$.

(c) Show that $\mathrm{Cl}\left(\mathcal{O}_{K}\right)$ is cyclic, and determine its order.

[You may assume that $\left.\mathcal{O}_{K}=\mathbb{Z}[\omega] .\right]$

Paper 4, Section II, 20G

comment(a) Let $L$ be a number field, and suppose there exists $\alpha \in \mathcal{O}_{L}$ such that $\mathcal{O}_{L}=\mathbb{Z}[\alpha]$. Let $f(X) \in \mathbb{Z}[X]$ denote the minimal polynomial of $\alpha$, and let $p$ be a prime. Let $\bar{f}(X) \in(\mathbb{Z} / p \mathbb{Z})[X]$ denote the reduction modulo $p$ of $f(X)$, and let

$\bar{f}(X)=\bar{g}_{1}(X)^{e_{1}} \cdots \bar{g}_{r}(X)^{e_{r}}$

denote the factorisation of $\bar{f}(X)$ in $(\mathbb{Z} / p \mathbb{Z})[X]$ as a product of powers of distinct monic irreducible polynomials $\bar{g}_{1}(X), \ldots, \bar{g}_{r}(X)$, where $e_{1}, \ldots, e_{r}$ are all positive integers.

For each $i=1, \ldots, r$, let $g_{i}(X) \in \mathbb{Z}[X]$ be any polynomial with reduction modulo $p$ equal to $\bar{g}_{i}(X)$, and let $P_{i}=\left(p, g_{i}(\alpha)\right) \subset \mathcal{O}_{L}$. Show that $P_{1}, \ldots, P_{r}$ are distinct, non-zero prime ideals of $\mathcal{O}_{L}$, and that there is a factorisation

$p \mathcal{O}_{L}=P_{1}^{e_{1}} \cdots P_{r}^{e_{r}}$

and that $N\left(P_{i}\right)=p^{\operatorname{deg} \bar{g}_{i}(X)}$.

(b) Let $K$ be a number field of degree $n=[K: \mathbb{Q}]$, and let $p$ be a prime. Suppose that there is a factorisation

$p \mathcal{O}_{K}=Q_{1} \cdots Q_{s}$

where $Q_{1}, \ldots, Q_{s}$ are distinct, non-zero prime ideals of $\mathcal{O}_{K}$ with $N\left(Q_{i}\right)=p$ for each $i=$ $1, \ldots, s$. Use the result of part (a) to show that if $n>p$ then there is no $\alpha \in \mathcal{O}_{K}$ such that $\mathcal{O}_{K}=\mathbb{Z}[\alpha]$.

Paper 1, Section II, G

comment(a) Let $m \geqslant 2$ be an integer such that $p=4 m-1$ is prime. Suppose that the ideal class group of $L=\mathbb{Q}(\sqrt{-p})$ is trivial. Show that if $n \geqslant 0$ is an integer and $n^{2}+n+m<m^{2}$, then $n^{2}+n+m$ is prime.

(b) Show that the ideal class group of $\mathbb{Q}(\sqrt{-163})$ is trivial.

Paper 2, Section II, 20G

commentLet $p \equiv 1 \bmod 4$ be a prime, and let $\omega=e^{2 \pi i / p}$. Let $L=\mathbb{Q}(\omega)$.

(a) Show that $[L: \mathbb{Q}]=p-1$.

(b) Calculate $\operatorname{disc}\left(1, \omega, \omega^{2}, \ldots, \omega^{p-2}\right)$. Deduce that $\sqrt{p} \in L$.

(c) Now suppose $p=5$. Prove that $\mathcal{O}_{L}^{\times}=\left\{\pm \omega^{a}\left(\frac{1}{2}+\frac{\sqrt{5}}{2}\right)^{b} \mid a, b \in \mathbb{Z}\right\}$. [You may use any general result without proof, provided that you state it precisely.]

Paper 4, Section II, G

commentLet $m \geqslant 2$ be a square-free integer, and let $n \geqslant 2$ be an integer. Let $L=\mathbb{Q}(\sqrt[n]{m})$.

(a) By considering the factorisation of $(m)$ into prime ideals, show that $[L: \mathbb{Q}]=n$.

(b) Let $T: L \times L \rightarrow \mathbb{Q}$ be the bilinear form defined by $T(x, y)=\operatorname{tr}_{L / \mathbb{Q}}(x y)$. Let $\beta_{i}=\sqrt[n]{m} i, i=0, \ldots, n-1$. Calculate the dual basis $\beta_{0}^{*}, \ldots, \beta_{n-1}^{*}$ of $L$ with respect to $T$, and deduce that $\mathcal{O}_{L} \subset \frac{1}{n m} \mathbb{Z}[\sqrt[n]{m}]$.

(c) Show that if $p$ is a prime and $n=m=p$, then $\mathcal{O}_{L}=\mathbb{Z}[\sqrt[p]{p}]$.

Paper 1, Section II, H

commentLet $\mathcal{O}_{L}$ be the ring of integers in a number field $L$, and let $\mathfrak{a} \leqslant \mathcal{O}_{L}$ be a non-zero ideal of $\mathcal{O}_{L}$.

(a) Show that $\mathfrak{a} \cap \mathbb{Z} \neq\{0\}$.

(b) Show that $\mathcal{O}_{L} / \mathfrak{a}$ is a finite abelian group.

(c) Show that if $x \in L$ has $x \mathfrak{a} \subseteq \mathfrak{a}$, then $x \in \mathcal{O}_{L}$.

(d) Suppose $[L: \mathbb{Q}]=2$, and $\mathfrak{a}=\langle b, \alpha\rangle$, with $b \in \mathbb{Z}$ and $\alpha \in \mathcal{O}_{L}$. Show that $\langle b, \alpha\rangle\langle b, \bar{\alpha}\rangle$ is principal.

[You may assume that $\mathfrak{a}$ has an integral basis.]

Paper 2, Section II, 18H

comment(a) Let $L$ be a number field, $\mathcal{O}_{L}$ the ring of integers in $L, \mathcal{O}_{L}^{*}$ the units in $\mathcal{O}_{L}, r$ the number of real embeddings of $L$, and $s$ the number of pairs of complex embeddings of $L$.

Define a group homomorphism $\mathcal{O}_{L}^{*} \rightarrow \mathbb{R}^{r+s-1}$ with finite kernel, and prove that the image is a discrete subgroup of $\mathbb{R}^{r+s-1}$.

(b) Let $K=\mathbb{Q}(\sqrt{d})$ where $d>1$ is a square-free integer. What is the structure of the group of units of $K$ ? Show that if $d$ is divisible by a prime $p \equiv 3(\bmod 4)$ then every unit of $K$ has norm $+1$. Find an example of $K$ with a unit of norm $-1$.

Paper 4, Section II, H

comment(a) Write down $\mathcal{O}_{K}$, when $K=\mathbb{Q}(\sqrt{\delta})$, and $\delta \equiv 2$ or $3(\bmod 4)$. [You need not prove your answer.]

Let $L=\mathbb{Q}(\sqrt{2}, \sqrt{\delta})$, where $\delta \equiv 3(\bmod 4)$ is a square-free integer. Find an integral basis of $\mathcal{O}_{L} \cdot$ [Hint: Begin by considering the relative traces $t r_{L / K}$, for $K$ a quadratic subfield of $L .]$

(b) Compute the ideal class group of $\mathbb{Q}(\sqrt{-14})$.

Paper 1, Section II, F

comment(a) Let $f(X) \in \mathbb{Q}[X]$ be an irreducible polynomial of degree $n, \theta \in \mathbb{C}$ a root of $f$, and $K=\mathbb{Q}(\theta)$. Show that $\operatorname{disc}(f)=\pm N_{K / \mathbb{Q}}\left(f^{\prime}(\theta)\right)$.

(b) Now suppose $f(X)=X^{n}+a X+b$. Write down the matrix representing multiplication by $f^{\prime}(\theta)$ with respect to the basis $1, \theta, \ldots, \theta^{n-1}$ for $K$. Hence show that

$\operatorname{disc}(f)=\pm\left((1-n)^{n-1} a^{n}+n^{n} b^{n-1}\right)$

(c) Suppose $f(X)=X^{4}+X+1$. Determine $\mathcal{O}_{K}$. [You may quote any standard result, as long as you state it clearly.]

Paper 2, Section II, F

comment(a) Prove that $5+2 \sqrt{6}$ is a fundamental unit in $\mathbb{Q}(\sqrt{6})$. [You may not assume the continued fraction algorithm.]

(b) Determine the ideal class group of $\mathbb{Q}(\sqrt{-55})$.

Paper 4, Section II, F

commentLet $K$ be a number field, and $p$ a prime in $\mathbb{Z}$. Explain what it means for $p$ to be inert, to split completely, and to be ramified in $K$.

(a) Show that if $[K: \mathbb{Q}]>2$ and $\mathcal{O}_{K}=\mathbb{Z}[\alpha]$ for some $\alpha \in K$, then 2 does not split completely in $K$.

(b) Let $K=\mathbb{Q}(\sqrt{d})$, with $d \neq 0,1$ and $d$ square-free. Determine, in terms of $d$, whether $p=2$ splits completely, is inert, or ramifies in $K$. Hence show that the primes which ramify in $K$ are exactly those which divide $D_{K}$.

Paper 1, Section II, $16 \mathrm{H}$

comment(a) Let $K$ be a number field, and $f$ a monic polynomial whose coefficients are in $\mathcal{O}_{K}$. Let $M$ be a field containing $K$ and $\alpha \in M$. Show that if $f(\alpha)=0$, then $\alpha$ is an algebraic integer.

Hence conclude that if $h \in K[x]$ is monic, with $h^{n} \in \mathcal{O}_{K}[x]$, then $h \in \mathcal{O}_{K}[x]$.

(b) Compute an integral basis for $\mathcal{O}_{\mathbb{Q}(\alpha)}$ when the minimum polynomial of $\alpha$ is $x^{3}-x-4$.

Paper 2, Section II, H

comment(i) Let $d \equiv 2$ or $3 \bmod 4$. Show that $(p)$ remains prime in $\mathcal{O}_{\mathbb{Q}(\sqrt{d})}$ if and only if $x^{2}-d$ is irreducible $\bmod p$.

(ii) Factorise $(2)$, (3) in $\mathcal{O}_{K}$, when $K=\mathbb{Q}(\sqrt{-14})$. Compute the class group of $K$.

Paper 4, Section II, H

commentLet $K$ be a number field. State Dirichlet's unit theorem, defining all the terms you use, and what it implies for a quadratic field $\mathbb{Q}(\sqrt{d})$, where $d \neq 0,1$ is a square-free integer.

Find a fundamental unit of $\mathbb{Q}(\sqrt{26})$.

Find all integral solutions $(x, y)$ of the equation $x^{2}-26 y^{2}=\pm 10$.

Paper 1, Section II, F

commentState a result involving the discriminant of a number field that implies that the class group is finite.

Use Dedekind's theorem to factor $2,3,5$ and 7 into prime ideals in $K=\mathbb{Q}(\sqrt{-34})$. By factoring $1+\sqrt{-34}$ and $4+\sqrt{-34}$, or otherwise, prove that the class group of $K$ is cyclic, and determine its order.

Paper 2, Section II, F

comment(i) Show that each prime ideal in a number field $K$ divides a unique rational prime $p$. Define the ramification index and residue class degree of such an ideal. State and prove a formula relating these numbers, for all prime ideals dividing a given rational prime $p$, to the degree of $K$ over $\mathbb{Q}$.

(ii) Show that if $\zeta_{n}$ is a primitive $n$th root of unity then $\prod_{j=1}^{n-1}\left(1-\zeta_{n}^{j}\right)=n$. Deduce that if $n=p q$, where $p$ and $q$ are distinct primes, then $1-\zeta_{n}$ is a unit in $\mathbb{Z}\left[\zeta_{n}\right]$.

(iii) Show that if $K=\mathbb{Q}\left(\zeta_{p}\right)$ where $p$ is prime, then any prime ideal of $K$ dividing $p$ has ramification index at least $p-1$. Deduce that $[K: \mathbb{Q}]=p-1$.

Paper 4, Section II, F

commentExplain what is meant by an integral basis for a number field. Splitting into the cases $d \equiv 1(\bmod 4)$ and $d \equiv 2,3(\bmod 4)$, find an integral basis for $K=\mathbb{Q}(\sqrt{d})$ where $d \neq 0,1$ is a square-free integer. Justify your answer.

Find the fundamental unit in $\mathbb{Q}(\sqrt{13})$. Determine all integer solutions to the equation $x^{2}+x y-3 y^{2}=17$.

Paper 1, Section II, H

commentLet $f \in \mathbb{Z}[X]$ be a monic irreducible polynomial of degree $n$. Let $K=\mathbb{Q}(\alpha)$, where $\alpha$ is a root of $f$.

(i) Show that if $\operatorname{disc}(f)$ is square-free then $\mathcal{O}_{K}=\mathbb{Z}[\alpha]$.

(ii) In the case $f(X)=X^{3}-3 X-25$ find the minimal polynomial of $\beta=3 /(1-\alpha)$ and hence compute the discriminant of $K$. What is the index of $\mathbb{Z}[\alpha]$ in $\mathcal{O}_{K}$ ?

[Recall that the discriminant of $X^{3}+p X+q$ is $-4 p^{3}-27 q^{2}$.]

Paper 2, Section II, H

comment(i) State Dirichlet's unit theorem.

(ii) Let $K$ be a number field. Show that if every conjugate of $\alpha \in \mathcal{O}_{K}$ has absolute value at most 1 then $\alpha$ is either zero or a root of unity.

(iii) Let $k=\mathbb{Q}(\sqrt{3})$ and $K=\mathbb{Q}(\zeta)$ where $\zeta=e^{i \pi / 6}=(i+\sqrt{3}) / 2$. Compute $N_{K / k}(1+\zeta)$. Show that

$\mathcal{O}_{K}^{*}=\left\{(1+\zeta)^{m} u: 0 \leqslant m \leqslant 11, u \in \mathcal{O}_{k}^{*}\right\}$

Hence or otherwise find fundamental units for $k$ and $K$.

[You may assume that the only roots of unity in $K$ are powers of $\zeta .$ ]

Paper 4, Section II, H

commentState Dedekind's criterion. Use it to factor the primes up to 5 in the ring of integers $\mathcal{O}_{K}$ of $K=\mathbb{Q}(\sqrt{65})$. Show that every ideal in $\mathcal{O}_{K}$ of norm 10 is principal, and compute the class group of $K$.

Paper 1, Section II, F

commentLet $K$ be a number field, and $\mathcal{O}_{K}$ its ring of integers. Write down a characterisation of the units in $\mathcal{O}_{K}$ in terms of the norm. Without assuming Dirichlet's units theorem, prove that for $K$ a quadratic field the quotient of the unit group by $\{\pm 1\}$ is cyclic (i.e. generated by one element). Find a generator in the cases $K=\mathbb{Q}(\sqrt{-3})$ and $K=\mathbb{Q}(\sqrt{11})$.

Determine all integer solutions of the equation $x^{2}-11 y^{2}=n$ for $n=-1,5,14$.

Paper 2, Section II, F

commentLet $K=\mathbb{Q}(\alpha)$ where $\alpha$ is a root of $X^{2}-X+12=0$. Factor the elements 2,3 , $\alpha$ and $\alpha+2$ as products of prime ideals in $\mathcal{O}_{K}$. Hence compute the class group of $K$.

Show that the equation $y^{2}+y=3\left(x^{5}-4\right)$ has no integer solutions.

Paper 4, Section II, F

commentLet $K=\mathbb{Q}(\sqrt{p}, \sqrt{q})$ where $p$ and $q$ are distinct primes with $p \equiv q \equiv 3(\bmod 4)$. By computing the relative traces $\operatorname{Tr}_{K / k}(\theta)$ where $k$ runs through the three quadratic subfields of $K$, show that the algebraic integers $\theta$ in $K$ have the form

$\theta=\frac{1}{2}(a+b \sqrt{p})+\frac{1}{2}(c+d \sqrt{p}) \sqrt{q}$

where $a, b, c, d$ are rational integers. Show further that if $c$ and $d$ are both even then $a$ and $b$ are both even. Hence prove that an integral basis for $K$ is

$1, \sqrt{p}, \frac{1+\sqrt{p q}}{2}, \frac{\sqrt{p}+\sqrt{q}}{2} .$

Calculate the discriminant of $K$.

Paper 1, Section II, F

commentCalculate the class group for the field $K=\mathbb{Q}(\sqrt{-17})$.

[You may use any general theorem, provided that you state it accurately.]

Find all solutions in $\mathbb{Z}$ of the equation $y^{2}=x^{5}-17$.

Paper 2, Section II, F

comment(i) Suppose that $d>1$ is a square-free integer. Describe, with justification, the ring of integers in the field $K=\mathbb{Q}(\sqrt{d})$.

(ii) Show that $\mathbb{Q}\left(2^{1 / 3}\right)=\mathbb{Q}\left(4^{1 / 3}\right)$ and that $\mathbb{Z}\left[4^{1 / 3}\right]$ is not the ring of integers in this field.

Paper 4, Section II, F

comment(i) Prove that the ring of integers $\mathcal{O}_{K}$ in a real quadratic field $K$ contains a non-trivial unit. Any general results about lattices and convex bodies may be assumed.

(ii) State the general version of Dirichlet's unit theorem.

(iii) Show that for $K=\mathbb{Q}(\sqrt{7}), 8+3 \sqrt{7}$ is a fundamental unit in $\mathcal{O}_{K}$.

[You may not use results about continued fractions unless you prove them.]

Paper 1, Section II, G

commentSuppose that $m$ is a square-free positive integer, $m \geqslant 5, m \not \equiv 1 \quad(\bmod 4)$. Show that, if the class number of $K=\mathbb{Q}(\sqrt{-m})$ is prime to 3 , then $x^{3}=y^{2}+m$ has at most two solutions in integers. Assume the $m$ is even.

Paper 2, Section II, G

commentCalculate the class group of the field $\mathbb{Q}(\sqrt{-14})$.

Paper 4, Section II, G

commentSuppose that $\alpha$ is a zero of $x^{3}-x+3$ and that $K=\mathbb{Q}(\alpha)$. Show that $[K: \mathbb{Q}]=3$. Show that $O_{K}$, the ring of integers in $K$, is $O_{K}=\mathbb{Z}[\alpha]$.

[You may quote any general theorem that you wish, provided that you state it clearly. Note that the discriminant of $x^{3}+p x+q$ is $-4 p^{3}-27 q^{2}$.]

Paper 1, Section II, H

commentSuppose that $K$ is a number field with ring of integers $\mathcal{O}_{K}$.

(i) Suppose that $M$ is a sub- $\mathbb{Z}$-module of $\mathcal{O}_{K}$ of finite index $r$ and that $M$ is closed under multiplication. Define the discriminant of $M$ and of $\mathcal{O}_{K}$, and show that $\operatorname{disc}(M)=r^{2} \operatorname{disc}\left(\mathcal{O}_{K}\right) .$

(ii) Describe $\mathcal{O}_{K}$ when $K=\mathbb{Q}[X] /\left(X^{3}+2 X+1\right)$.

[You may assume that the polynomial $X^{3}+p X+q$ has discriminant $-4 p^{3}-27 q^{2}$.]

(iii) Suppose that $f, g \in \mathbb{Z}[X]$ are monic quadratic polynomials with equal discriminant $d$, and that $d \notin\{0,1\}$ is square-free. Show that $\mathbb{Z}[X] /(f)$ is isomorphic to $\mathbb{Z}[X] /(g)$.

[Hint: Classify quadratic fields in terms of discriminants.]

Paper 2, Section II, H

commentSuppose that $K$ is a number field of degree $n=r+2 s$, where $K$ has exactly real embeddings.

(i) Taking for granted the fact that there is a constant $C_{K}$ such that every integral ideal $I$ of $\mathcal{O}_{K}$ has a non-zero element $x$ such that $|N(x)| \leqslant C_{K} N(I)$, deduce that the class group of $K$ is finite.

(ii) Compute the class group of $\mathbb{Q}(\sqrt{-21})$, given that you can take

$C_{K}=\left(\frac{4}{\pi}\right)^{s} \frac{n !}{n^{n}}\left|D_{K}\right|^{1 / 2}$

where $D_{K}$ is the discriminant of $K$.

(iii) Find all integer solutions of $y^{2}=x^{3}-21$.

Paper 4, Section II, H

commentSuppose that $K$ is a number field of degree $n=r+2 s$, where $K$ has exactly real embeddings.

Show that the group of units in $\mathcal{O}_{K}$ is a finitely generated abelian group of rank at most $r+s-1$. Identify the torsion subgroup in terms of roots of unity.

[General results about discrete subgroups of a Euclidean real vector space may be used without proof, provided that they are stated clearly.]

Find all the roots of unity in $\mathbb{Q}(\sqrt{11})$.

1.II.20G

comment(a) Define the ideal class group of an algebraic number field $K$. State a result involving the discriminant of $K$ that implies that the ideal class group is finite.

(b) Put $K=\mathbb{Q}(\omega)$, where $\omega=\frac{1}{2}(1+\sqrt{-23})$, and let $\mathcal{O}_{K}$ be the ring of integers of $K$. Show that $\mathcal{O}_{K}=\mathbb{Z}+\mathbb{Z} \omega$. Factorise the ideals [2] and [3] in $\mathcal{O}_{K}$ into prime ideals. Verify that the equation of ideals

$[2, \omega][3, \omega]=[\omega]$

holds. Hence prove that $K$ has class number 3 .

2.II.20G

comment(a) Factorise the ideals [2], [3] and [5] in the ring of integers $\mathcal{O}_{K}$ of the field $K=\mathbb{Q}(\sqrt{30})$. Using Minkowski's bound

$\frac{n !}{n^{n}}\left(\frac{4}{\pi}\right)^{s} \sqrt{\left|d_{K}\right|},$

determine the ideal class group of $K$.

[Hint: it might be helpful to notice that $\frac{3}{2}=N_{K / \mathbb{Q}}(\alpha)$ for some $\left.\alpha \in K .\right]$

(b) Find the fundamental unit of $K$ and determine all solutions of the equations $x^{2}-30 y^{2}=\pm 5$ in integers $x, y \in \mathbb{Z}$. Prove that there are in fact no solutions of $x^{2}-30 y^{2}=5$ in integers $x, y \in \mathbb{Z}$.

4.II.20G

comment(a) Explain what is meant by an integral basis of an algebraic number field. Specify such a basis for the quadratic field $k=\mathbb{Q}(\sqrt{2})$.

(b) Let $K=\mathbb{Q}(\alpha)$ with $\alpha=\sqrt[4]{2}$, a fourth root of 2 . Write an element $\theta$ of $K$ as

$\theta=a+b \alpha+c \alpha^{2}+d \alpha^{3}$

with $a, b, c, d \in \mathbb{Q}$. By computing the relative traces $T_{K / k}(\theta)$ and $T_{K / k}(\alpha \theta)$, show that if $\theta$ is an algebraic integer of $K$, then $2 a, 2 b, 2 c$ and $4 d$ are rational integers. By further computing the relative norm $N_{K / k}(\theta)$, show that

$a^{2}+2 c^{2}-4 b d \text { and } 2 a c-b^{2}-2 d^{2}$

are rational integers. Deduce that $1, \alpha, \alpha^{2}, \alpha^{3}$ is an integral basis of $K$.

1.II.20H

commentLet $K=\mathbb{Q}(\sqrt{-26})$.

(a) Show that $\mathcal{O}_{K}=\mathbb{Z}[\sqrt{-26}]$ and that the discriminant $d_{K}$ is equal to $-104$.

(b) Show that 2 ramifies in $\mathcal{O}_{K}$ by showing that $[2]=\mathfrak{p}_{2}^{2}$, and that $\mathfrak{p}_{2}$ is not a principal ideal. Show further that $[3]=\mathfrak{p}_{3} \overline{\mathfrak{p}}_{3}$ with $\mathfrak{p}_{3}=[3,1-\sqrt{-26}]$. Deduce that neither $\mathfrak{p}_{3}$ nor $\mathfrak{p}_{3}^{2}$ is a principal ideal, but $\mathfrak{p}_{3}^{3}=[1-\sqrt{-26}]$.

(c) Show that 5 splits in $\mathcal{O}_{K}$ by showing that $[5]=\mathfrak{p}_{5} \overline{\mathfrak{p}}_{5}$, and that

$N_{K / \mathbb{Q}}(2+\sqrt{-26})=30$

Deduce that $\mathfrak{p}_{2} \mathfrak{p}_{3} \mathfrak{p}_{5}$ has trivial class in the ideal class group of $K$. Conclude that the ideal class group of $K$ is cyclic of order six.

[You may use the fact that $\left.\frac{2}{\pi} \sqrt{104} \approx 6.492 .\right]$

2.II.20H

commentLet $K=\mathbb{Q}(\sqrt{10})$ and put $\varepsilon=3+\sqrt{10}$.

(a) Show that 2,3 and $\varepsilon+1$ are irreducible elements in $\mathcal{O}_{K}$. Deduce from the equation

$6=2 \cdot 3=(\varepsilon+1)(\bar{\varepsilon}+1)$

that $\mathcal{O}_{K}$ is not a principal ideal domain.

(b) Put $\mathfrak{p}_{2}=[2, \varepsilon+1]$ and $\mathfrak{p}_{3}=[3, \varepsilon+1]$. Show that

$[2]=\mathfrak{p}_{2}^{2}, \quad[3]=\mathfrak{p}_{3} \overline{\mathfrak{p}}_{3}, \quad \mathfrak{p}_{2} \mathfrak{p}_{3}=[\varepsilon+1], \quad \mathfrak{p}_{2} \overline{\mathfrak{p}}_{3}=[\varepsilon-1] .$

Deduce that $K$ has class number 2 .

(c) Show that $\varepsilon$ is the fundamental unit of $K$. Hence prove that all solutions in integers $x, y$ of the equation $x^{2}-10 y^{2}=6$ are given by

$x+\sqrt{10} y=\pm \varepsilon^{n}\left(\varepsilon+(-1)^{n}\right), \quad n=0,1,2, \ldots$

4.II.20H

commentLet $K$ be a finite extension of $\mathbb{Q}$ and let $\mathcal{O}=\mathcal{O}_{K}$ be its ring of integers. We will assume that $\mathcal{O}=\mathbb{Z}[\theta]$ for some $\theta \in \mathcal{O}$. The minimal polynomial of $\theta$ will be denoted by $g$. For a prime number $p$ let

$\bar{g}(X)=\bar{g}_{1}(X)^{e_{1}} \cdot \ldots \cdot \bar{g}_{r}(X)^{e_{r}}$

be the decomposition of $\bar{g}(X)=g(X)+p \mathbb{Z}[X] \in(\mathbb{Z} / p \mathbb{Z})[X]$ into distinct irreducible monic factors $\bar{g}_{i}(X) \in(\mathbb{Z} / p \mathbb{Z})[X]$. Let $g_{i}(X) \in \mathbb{Z}[X]$ be a polynomial whose reduction modulo $p$ is $\bar{g}_{i}(X)$. Show that

$\mathfrak{p}_{i}=\left[p, g_{i}(\theta)\right], \quad i=1, \ldots, r,$

are the prime ideals of $\mathcal{O}$ containing $p$, that these are pairwise different, and

$[p]=\mathfrak{p}_{1}^{e_{1}} \cdot \ldots \cdot \mathfrak{p}_{r}^{e_{r}}$

1.II.20G

commentLet $\alpha, \beta, \gamma$ denote the zeros of the polynomial $x^{3}-n x-1$, where $n$ is an integer. The discriminant of the polynomial is defined as

$\Delta=\Delta\left(1, \alpha, \alpha^{2}\right)=(\alpha-\beta)^{2}(\beta-\gamma)^{2}(\gamma-\alpha)^{2}$

Prove that, if $\Delta$ is square-free, then $1, \alpha, \alpha^{2}$ is an integral basis for $k=\mathbb{Q}(\alpha)$.

By verifying that

$\alpha(\alpha-\beta)(\alpha-\gamma)=2 n \alpha+3$

and further that the field norm of the expression on the left is $-\Delta$, or otherwise, show that $\Delta=4 n^{3}-27$. Hence prove that, when $n=1$ and $n=2$, an integral basis for $k$ is $1, \alpha, \alpha^{2}$.

2.II.20G

commentLet $K=\mathbb{Q}(\sqrt{26})$ and let $\varepsilon=5+\sqrt{26}$. By Dedekind's theorem, or otherwise, show that the ideal equations

$2=[2, \varepsilon+1]^{2}, \quad 5=[5, \varepsilon+1][5, \varepsilon-1], \quad[\varepsilon+1]=[2, \varepsilon+1][5, \varepsilon+1]$

hold in $K$. Deduce that $K$ has class number 2 .

Show that $\varepsilon$ is the fundamental unit in $K$. Hence verify that all solutions in integers $x, y$ of the equation $x^{2}-26 y^{2}=\pm 10$ are given by

$x+\sqrt{26} y=\pm \varepsilon^{n}(\varepsilon \pm 1) \quad(n=0, \pm 1, \pm 2, \ldots) .$

[It may be assumed that the Minkowski constant for $K$ is $\frac{1}{2}$.]

4.II.20G

commentLet $\zeta=e^{2 \pi i / 5}$ and let $K=\mathbb{Q}(\zeta)$. Show that the discriminant of $K$ is 125 . Hence prove that the ideals in $K$ are all principal.

Verify that $\left(1-\zeta^{n}\right) /(1-\zeta)$ is a unit in $K$ for each integer $n$ with $1 \leqslant n \leqslant 4$. Deduce that $5 /(1-\zeta)^{4}$ is a unit in $K$. Hence show that the ideal $[1-\zeta]$ is prime and totally ramified in $K$. Indicate briefly why there are no other ramified prime ideals in $K$.

[It can be assumed that $\zeta, \zeta^{2}, \zeta^{3}, \zeta^{4}$ is an integral basis for $K$ and that the Minkowski constant for $K$ is $3 /\left(2 \pi^{2}\right)$.]

1.II.20G

commentLet $K=\mathbb{Q}(\sqrt{2}, \sqrt{p})$ where $p$ is a prime with $p \equiv 3 \operatorname{(\operatorname {mod}4)\text {.Bycomputingthe}}$ relative traces $\operatorname{Tr}_{K / k}(\theta)$ where $k$ runs through the three quadratic subfields of $K$, show that the algebraic integers $\theta$ in $K$ have the form

$\theta=\frac{1}{2}(a+b \sqrt{p})+\frac{1}{2}(c+d \sqrt{p}) \sqrt{2}$

where $a, b, c, d$ are rational integers. By further computing the relative norm $\mathrm{N}_{K / k}(\theta)$ where $k=\mathbb{Q}(\sqrt{2})$, show that 4 divides

$a^{2}+p b^{2}-2\left(c^{2}+p d^{2}\right) \quad \text { and } \quad 2(a b-2 c d)$

Deduce that $a$ and $b$ are even and $c \equiv d(\bmod 2)$. Hence verify that an integral basis for $K$ is

$1, \quad \sqrt{2}, \quad \sqrt{p}, \quad \frac{1}{2}(1+\sqrt{p}) \sqrt{2} .$

2.II.20G

commentShow that $\varepsilon=(3+\sqrt{7}) /(3-\sqrt{7})$ is a unit in $k=\mathbb{Q}(\sqrt{7})$. Show further that 2 is the square of the principal ideal in $k$ generated by $3+\sqrt{7}$.

Assuming that the Minkowski constant for $k$ is $\frac{1}{2}$, deduce that $k$ has class number 1 .

Assuming further that $\varepsilon$ is the fundamental unit in $k$, show that the complete solution in integers $x, y$ of the equation $x^{2}-7 y^{2}=2$ is given by

$x+\sqrt{7} y=\pm \varepsilon^{n}(3+\sqrt{7}) \quad(n=0, \pm 1, \pm 2, \ldots) .$

Calculate the particular solution in positive integers $x, y$ when $n=1$

4.II.20G

commentState Dedekind's theorem on the factorisation of rational primes into prime ideals.

A rational prime is said to ramify totally in a field with degree $n$ if it is the $n$-th power of a prime ideal in the field. Show that, in the quadratic field $\mathbb{Q}(\sqrt{d})$ with $d$ a squarefree integer, a rational prime ramifies totally if and only if it divides the discriminant of the field.

Verify that the same holds in the cyclotomic field $\mathbb{Q}(\zeta)$, where $\zeta=e^{2 \pi i / q}$ with $q$ an odd prime, and also in the cubic field $\mathbb{Q}(\sqrt[3]{2})$.

$[$ The cases $d \equiv 2,3(\bmod 4)$ and $d \equiv 1(\bmod 4)$ for the quadratic field should be carefully distinguished. It can be assumed that $\mathbb{Q}(\zeta)$ has a basis $1, \zeta, \ldots, \zeta^{q-2}$ and discriminant $(-1)^{(q-1) / 2} q^{q-1}$, and that $\mathbb{Q}(\sqrt[3]{2})$ has a basis $1, \sqrt[3]{2},(\sqrt[3]{2})^{2}$ and discriminant $\left.-108 .\right]$

B1.9

commentLet $K=\mathbb{Q}(\theta)$, where $\theta$ is a root of $X^{3}-4 X+1$. Prove that $K$ has degree 3 over $\mathbb{Q}$, and admits three distinct embeddings in $\mathbb{R}$. Find the discriminant of $K$ and determine the ring of integers $\mathcal{O}$ of $K$. Factorise $2 \mathcal{O}$ and $3 \mathcal{O}$ into a product of prime ideals.

Using Minkowski's bound, show that $K$ has class number 1 provided all prime ideals in $\mathcal{O}$ dividing 2 and 3 are principal. Hence prove that $K$ has class number $1 .$

[You may assume that the discriminant of $X^{3}+a X+b$ is $-4 a^{3}-27 b^{2}$.]

B2.9

commentLet $m$ be an integer greater than 1 and let $\zeta_{m}$ denote a primitive $m$-th root of unity in $\mathbb{C}$. Let $\mathcal{O}$ be the ring of integers of $\mathbb{Q}\left(\zeta_{m}\right)$. If $p$ is a prime number with $(p, m)=1$, outline the proof that

$p \mathcal{O}=\wp_{1} \cdots \wp_{r},$

where the $\wp_{i}$ are distinct prime ideals of $\mathcal{O}$, and $r=\varphi(m) / f$ with $f$ the least integer $\geqslant 1$ such that $p^{f} \equiv 1 \bmod m$. [Here $\varphi(m)$ is the Euler $\varphi$-function of $\left.m\right]$.

Determine the factorisations of $2,3,5$ and 11 in $\mathbb{Q}\left(\zeta_{5}\right)$. For each integer $n \geqslant 1$, prove that, in the ring of integers of $\mathbb{Q}\left(\zeta_{5^{n}}\right)$, there is a unique prime ideal dividing 2 , and a unique prime ideal dividing 3 .

B4.6

commentLet $K$ be a finite extension of $\mathbb{Q}$, and $\mathcal{O}$ the ring of integers of $K$. Write an essay outlining the proof that every non-zero ideal of $\mathcal{O}$ can be written as a product of non-zero prime ideals, and that this factorisation is unique up to the order of the factors.

B1.9

commentLet $K=\mathbb{Q}(\alpha)$, where $\alpha=\sqrt[3]{10}$, and let $\mathcal{O}_{K}$ be the ring of algebraic integers of $K$. Show that the field polynomial of $r+s \alpha$, with $r$ and $s$ rational, is $(x-r)^{3}-10 s^{3}$.

Let $\beta=\frac{1}{3}\left(\alpha^{2}+\alpha+1\right)$. By verifying that $\beta=3 /(\alpha-1)$ and determining the field polynomial, or otherwise, show that $\beta$ is in $\mathcal{O}_{K}$.

By computing the traces of $\theta, \alpha \theta, \alpha^{2} \theta$, show that the elements of $\mathcal{O}_{K}$ have the form

$\theta=\frac{1}{3}\left(l+\frac{1}{10} m \alpha+\frac{1}{10} n \alpha^{2}\right)$

where $l, m, n$ are integers. By further computing the norm of $\frac{1}{10} \alpha(m+n \alpha)$, show that $\theta$ can be expressed as $\frac{1}{3}(u+v \alpha)+w \beta$ with $u, v, w$ integers. Deduce that $1, \alpha, \beta$ form an integral basis for $K$.

B2.9

commentBy Dedekind's theorem, or otherwise, factorise $2,3,5$ and 7 into prime ideals in the field $K=\mathbb{Q}(\sqrt{-34})$. Show that the ideal equations

$[\omega]=[5, \omega][7, \omega], \quad[\omega+3]=[2, \omega+3][5, \omega+3]^{2}$

hold in $K$, where $\omega=1+\sqrt{-34}$. Hence, prove that the ideal class group of $K$ is cyclic of order $4 .$

[It may be assumed that the Minkowski constant for $K$ is $2 / \pi$.]

B4.6

commentWrite an essay on the Dirichlet unit theorem with particular reference to quadratic fields.

B1.9

Explain what is meant by an integral basis $\omega_{1}, \ldots, \omega_{n}$ of a number field $K$. Give an expression for the discriminant of $K$ in terms of the traces of the $\omega_{i} \omega_{j}$.

Let $K=\mathbb{Q}(i, \sqrt{2})$. By computing the traces $T_{K / k}(\theta)$, where $k$ runs through the three quadratic subfields of $K$, show that the algebraic integers $\theta$ in $K$ have the form $\frac{1}{2}(\alpha+\beta \sqrt{2})$, where $\alpha=a+i b$ and $\beta=c+i d$ are Gaussian integers. By further computing the norm $N_{K / k}(\theta)$, where $k=\mathbb{Q}(\sqrt{2})$, show that $a$ and $b$ are even and that $c \equiv d(\bmod 2)$. Hence prove that an integral basis for $K$ is $1, i, \sqrt{2}, \frac{1}{2}(1+i) \sqrt{2}$