Part II — Number Fields —

Year 2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2021 73

Paper 1, Section II 20G Number Fields Let K = Q(α), where α3 = 5α 8. − (a) Show that [K : Q] = 3. (b) Let β = (α + α2)/2. By considering the matrix of β acting on K by multiplication, or otherwise, show that β is an algebraic integer, and that (1, α, β) is 3 a Z-basis for K .[The discriminant of T 5T + 8 is 4 307, and 307 is prime.] O − − · (c) Compute the prime factorisation of the ideal (3) in . Is (2) a prime ideal of OK ? Justify your answer. OK

Paper 2, Section II 20G Number Fields Let K be a field containing Q. What does it mean to say that an element of K is algebraic? Show that if α K is algebraic and non-zero, then there exists β Z[α] such ∈ ∈ that αβ is a non-zero (rational) integer. Now let K be a number field, with ring of integers . Let R be a subring of OK OK whose field of fractions equals K. Show that every element of K can be written as r/m, where r R and m is a positive integer. ∈ Prove that R is a free abelian group of rank [K : Q], and that R has finite index in . Show also that for every nonzero ideal I of R, the index (R : I) of I in R is finite, OK and that for some positive integer m, m is an ideal of R. OK Suppose that for every pair of non-zero ideals I, J R, we have ⊂ (R : IJ) = (R : I)(R : J) .

Show that R = . OK [You may assume without proof that K is a free abelian group of rank [K : Q]. ] O

Part II, 2021 List of Questions [TURN OVER] 2021 74

Paper 4, Section II 20G Number Fields (a) Compute the class group of K = Q(√30). Find also the fundamental unit of K, stating clearly any general results you use. [The Minkowski bound for a real quadratic field is d 1/2/2. ] | K | (b) Let K = Q(√d) be real quadratic, with embeddings σ1, σ2 , R. An element → α K is totally positive if σ (α) > 0 and σ (α) > 0. Show that the totally positive ∈ 1 2 elements of K form a subgroup of the multiplicative group K∗ of index 4. Let I, J be non-zero ideals. We say that I is narrowly equivalent to J if ⊂ OK there exists a totally positive element α of K such that I = αJ. 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

the class number h of K if the fundamental unit of K has norm 1, K − (2hK otherwise.

Part II, 2021 List of Questions 2020 67

Paper 1, Section II 20G Number Fields State Minkowski’s theorem. Let K = Q(√ d), where d is a square-free positive integer, not congruent to 3 − (mod 4). Show that every nonzero ideal I contains an element α with ⊂ OK 4√d 0 < N (α) N(I). K/Q 6 π

Deduce the finiteness of the class group of K . Compute the class group of Q(√ 22). Hence show that the equation y3 = x2 + 22 − has no integer solutions.

Paper 2, Section II 20G Number Fields (a) Let K be a number field of degree n. Define the discriminant disc(α1, . . . , αn) of an n-tuple of elements αi of K, and show that it is nonzero if and only if α1, . . . , αn is a Q-basis for K. (b) Let K = Q(α) where α has minimal polynomial

n 1 n − j T + ajT , aj Z ∈ Xj=0 and assume that p is a prime such that, for every j, a 0 (mod p), but a 0 (mod p2). j ≡ 0 6≡ (i) Show that P = (p, α) is a prime ideal, that P n = (p) and that α / P 2. [Do not ∈ assume that K = Z[α].] O (ii) Show that the index of Z[α] in K is prime to p. O 3 (iii) If K = Q(α) with α + 3α + 3 = 0, show that K = Z[α]. [You may assume O without proof that the discriminant of T 3 + aT + b is 4a3 27b2.] − −

Paper 4, Section II 20G Number Fields Let K be a number field of degree n, and let σi : K, C be the set of complex { → } embeddings of K. Show that if α satisfies σ (α) = 1 for every i, then α is a root ∈ OK | i | of unity. Prove also that there exists c > 1 such that if α and σ (α) < c for all i, ∈ OK | i | then α is a root of unity. State Dirichlet’s Unit theorem. Let K 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 , and that the group of units of K is then n n Z . Determine  for Q(√11), justifying your result. [If {± | ∈ } you use the continued fraction algorithm, you must prove it in full.]

Part II, 2020 List of Questions [TURN OVER] 2019

68

Paper 4, Section II 20G Number Fields (a) Let L be a number field, and suppose there exists α L such that L = Z[α]. ∈ O O Let f(X) Z[X] denote the minimal polynomial of α, and let p be a prime. Let ∈ f(X) (Z/pZ)[X] denote the reduction modulo p of f(X), and let ∈ f(X)= g (X)e1 g (X)er 1 r denote the factorisation of f(X) in (Z/pZ)[X] as a product of powers of distinct monic irreducible polynomials g1(X),..., gr(X), where e1,...,er are all positive integers.

For each i = 1, . . . , r, let gi(X) Z[X] be any polynomial with reduction modulo p ∈ equal to g (X), and let P = (p, g (α)) . Show that P ,...,P are distinct, non-zero i i i ⊂ OL 1 r prime ideals of , and that there is a factorisation OL p = P e1 P er , OL 1 r deg g (X) and that N(Pi)= p i . (b) Let K be a number field of degree n = [K : Q], and let p be a prime. Suppose that there is a factorisation p = Q Q , OK 1 s where Q ,...,Q are distinct, non-zero prime ideals of with N(Q ) = p for each i = 1 s OK i 1,...,s. Use the result of part (a) to show that if n>p then there is no α such that ∈ OK K = Z[α]. O

Paper 2, Section II 20G Number Fields (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 Cl( ). OL (b) Now let K = Q(√ 47) and ω = 1 (1 + √ 47). Using Dedekind’s criterion, or − 2 − otherwise, factorise the ideals (ω) and (2+ ω) as products of non-zero prime ideals of . OK (c) Show that Cl( ) is cyclic, and determine its order. OK [You may assume that K = Z[ω].] O

Part II, 2019 List of Questions 2019

69

Paper 1, Section II 20G Number Fields Let K = Q(√2). (a) Write down the ring of integers . OK (b) State Dirichlet’s unit theorem, and use it to determine all elements of the group of units × . OK (c) Let P denote the ideal generated by 3 + √2. Show that the group ⊂ OK

G = α × α 1 mod P { ∈ OK | ≡ } is cyclic, and find a generator.

Part II, 2019 List of Questions [TURN OVER 2018

67

Paper 2, Section II 20G Number Fields Let p 1 mod 4 be a prime, and let ω = e2πi/p. Let L = Q(ω). ≡ (a) Show that [L : Q]= p 1. − 2 p 2 (b) Calculate disc(1,ω,ω ,...,ω − ). Deduce that √p L. ∈ a 1 √5 b (c) Now suppose p = 5. Prove that × = ω ( + ) a, b Z . [You may use OL {± 2 2 | ∈ } any general result without proof, provided that you state it precisely.]

Paper 4, Section II 20G Number Fields Let m 2 be a square-free integer, and let n 2 be an integer. Let L = Q( √n m). (a) By considering the factorisation of (m) into prime ideals, show that [L : Q]= n.

(b) Let T : L L Q be the bilinear form defined by T (x,y) = trL/Q(xy). Let n i × → βi = √m , i = 0,...,n 1. Calculate the dual basis β0∗,...,βn∗ 1 of L with respect to T , −1 n − and deduce that L Z[ √m ]. O ⊂ nm p (c) Show that if p is a prime and n = m = p, then L = Z[√p ]. O

Paper 1, Section II 20G Number Fields (a) Let m 2 be an integer such that p = 4m 1 is prime. Suppose that the ideal − class group of L = Q(√ p) is trivial. Show that if n 0 is an integer and n2 +n+m

Part II, 2018 List of Questions [TURN OVER 2017

68

Paper 2, Section II 18H Number Fields

(a) Let L be a number field, the ring of integers in L, the units in , r the OL OL∗ OL number of real embeddings of L, and s the number of pairs of complex embeddings of L. r+s 1 Define a group homomorphism L∗ R − with finite kernel, and prove that the Or+→s 1 image is a discrete subgroup of R − .

(b) Let K = Q(√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 3 (mod 4) then every ≡ unit of K has norm +1. Find an example of K with a unit of norm 1. −

Paper 1, Section II 19H Number Fields Let be the ring of integers in a number field L, and let a be a non-zero OL OL ideal of . OL

(a) Show that a Z = 0 . ∩ { } (b) Show that /a is a finite abelian group. OL (c) Show that if x L has xa a, then x . ∈ ⊆ ∈ OL (d) Suppose [L : Q] =2, and a = b, α , with b Z and α L. Show that b, α b, α ∈ ∈ O is principal.

[You may assume that a has an integral basis.]

Part II, 2017 List of Questions 2017

69

Paper 4, Section II 19H Number Fields

(a) Write down K , when K = Q(√δ), and δ 2 or 3 (mod 4). [You need not prove O ≡ your answer.] Let L = Q(√2, √δ), where δ 3 (mod 4) is a square-free integer. Find an integral ≡ basis of . [Hint: Begin by considering the relative traces tr , for K a quadratic OL L/K subfield of L.]

(b) Compute the ideal class group of Q(√ 14). −

Part II, 2017 List of Questions [TURN OVER 2016

62

Paper 2, Section II 18F Number Fields (a) Prove that 5 + 2√6 is a fundamental unit in Q(√6). [You may not assume the continued fraction algorithm.] (b) Determine the ideal class group of Q(√ 55). −

Paper 1, Section II 19F Number Fields (a) Let f(X) Q[X] be an irreducible polynomial of degree n, θ C a root of f, ∈ ∈ and K = Q(θ). Show that disc(f)= NK/ (f ′(θ)). ± Q (b) Now suppose f(X) = Xn + aX + b. Write down the matrix representing n 1 multiplication by f ′(θ) with respect to the basis 1,θ,...,θ − for K. Hence show that

n 1 n n n 1 disc(f)= (1 n) − a + n b − . ± − (c) Suppose f(X) = X4 + X + 1. Determine . [You may quote any standard OK result, as long as you state it clearly.]

Paper 4, Section II 19F Number Fields Let K be a number field, and p a prime in Z. Explain what it means for p to be inert, to split completely, and to be ramified in K.

(a) Show that if [K : Q] > 2 and K = Z[α] for some α K, then 2 does not split O ∈ completely in K. (b) Let K = Q(√d), with d = 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 DK .

Part II, 2016 List of Questions 2015

63

Paper 4, Section II 16H Number Fields Let K be a number field. State Dirichlet’s unit theorem, defining all the terms you use, and what it implies for a quadratic field Q(√d), where d = 0, 1 is a square-free integer. Find a fundamental unit of Q(√26). Find all integral solutions (x,y) of the equation x2 26y2 = 10. − ±

Paper 2, Section II 16H Number Fields (i) Let d 2 or 3 mod 4. Show that (p) remains prime in if and only if ≡ OQ(√d) x2 d is irreducible mod p. − (ii) Factorise (2), (3) in K, when K = Q(√ 14). Compute the class group of K. O −

Paper 1, Section II 16H Number Fields (a) Let K be a number field, and f a monic polynomial whose coefficients are in . Let M be a field containing K and α M. Show that if f(α) = 0, then α is an OK ∈ algebraic integer. Hence conclude that if h K[x] is monic, with hn [x], then h [x]. ∈ ∈ OK ∈ OK (b) Compute an integral basis for when the minimum polynomial of α is OQ(α) x3 x 4. − −

Part II, 2015 List of Questions [TURN OVER 2014

63

Paper 4, Section II 20F Number Fields Explain what is meant by an integral basis for a number field. Splitting into the cases d 1 (mod 4) and d 2, 3 (mod 4), find an integral basis for K = Q(√d) where ≡ ≡ d = 0, 1 is a square-free integer. Justify your answer. Find the fundamental unit in Q(√13). Determine all integer solutions to the equation x2 + xy 3y2 = 17. −

Paper 2, Section II 20F Number Fields (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 Q. n 1 j (ii) Show that if ζ is a primitive nth root of unity then − (1 ζn)= n. Deduce n j=1 − that if n = pq, where p and q are distinct primes, then 1 ζn is a unit in Z[ζn]. − Q (iii) Show that if K = Q(ζp) where p is prime, then any prime ideal of K dividing p has ramification index at least p 1. Deduce that [K : Q]= p 1. − −

Paper 1, Section II 20F Number Fields State 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 = Q(√ 34). − By factoring 1 + √ 34 and 4+ √ 34, or otherwise, prove that the class group of K is − − cyclic, and determine its order.

Part II, 2014 List of Questions [TURN OVER 2013

63

Paper 4, Section II 20H Number Fields State Dedekind’s criterion. Use it to factor the primes up to 5 in the ring of integers K of K = Q(√65). Show that every ideal in K of norm 10 is principal, and compute O O the class group of K.

Paper 2, Section II 20H Number Fields (i) State Dirichlet’s unit theorem.

(ii) Let K be a number field. Show that if every conjugate of α has absolute value ∈ OK at most 1 then α is either zero or a root of unity.

iπ/6 (iii) Let k = Q(√3) and K = Q(ζ) where ζ = e = (i + √3)/2. Compute NK/k(1 + ζ). Show that m ∗ = (1 + ζ) u : 0 m 11, u ∗ . OK { 6 6 ∈ Ok} Hence or otherwise find fundamental units for k and K. [You may assume that the only roots of unity in K are powers of ζ.]

Paper 1, Section II 20H Number Fields Let f Z[X] be a monic irreducible polynomial of degree n. Let K = Q(α), where ∈ α is a root of f.

(i) Show that if disc(f) is square-free then K = Z[α]. O (ii) In the case f(X)= X3 3X 25 find the minimal polynomial of β = 3/(1 α) and − − − hence compute the discriminant of K. What is the index of Z[α] in K ? O [Recall that the discriminant of X3 + pX + q is 4p3 27q2.] − −

Part II, 2013 List of Questions [TURN OVER 2012

63

Paper 4, Section II 20F Number Fields Let K = Q(√p, √q) where p and q are distinct primes with p q 3 (mod 4). By ≡ ≡ computing the relative traces TrK/k(θ) where k runs through the three quadratic subfields of K, show that the algebraic integers θ in K have the form 1 1 θ = (a + b√p)+ (c + d√p)√q , 2 2 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+ √pq √p + √q 1 , √p , , . 2 2 Calculate the discriminant of K.

Paper 2, Section II 20F Number Fields Let K = Q(α) where α is a root of X2 X + 12 = 0. Factor the elements 2, 3, α − and α + 2 as products of prime ideals in . Hence compute the class group of K. OK Show that the equation y2 + y = 3(x5 4) has no integer solutions. −

Paper 1, Section II 20F Number Fields Let K be a number field, and its ring of integers. Write down a characterisation OK of the units in in terms of the norm. Without assuming Dirichlet’s units theorem, OK prove that for K a quadratic field the quotient of the unit group by 1 is cyclic (i.e. {± } generated by one element). Find a generator in the cases K = Q(√ 3) and K = Q(√11). − Determine all integer solutions of the equation x2 11y2 = n for n = 1, 5, 14. − −

Part II, 2012 List of Questions [TURN OVER 2011

61

Paper 1, Section II 20F Number Fields Calculate the class group for the field K = Q(√ 17). − [You may use any general theorem, provided that you state it accurately.] Find all solutions in Z of the equation y2 = x5 17. −

Paper 2, Section II 20F Number Fields

(i) Suppose that d> 1 is a square-free integer. Describe, with justification, the ring of integers in the field K = Q(√d).

(ii) Show that Q(21/3)= Q(41/3) and that Z[41/3] is not the ring of integers in this field.

Paper 4, Section II 20F Number Fields

(i) Prove that the ring of integers in a real quadratic field K contains a non-trivial OK 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 = Q(√7), 8 + 3√7 is a fundamental unit in K . O [You may not use results about continued fractions unless you prove them.]

Part II, 2011 List of Questions [TURN OVER 2010

60

Paper 1, Section II 20G Number Fields Suppose that m is a square-free positive integer, m > 5 , m 1 (mod4). Show 3 ≡ 2 that, if the class number of K = Q(√ m ) is prime to 3, then x = y + m has at most − two solutions in integers. Assume the m is even.

Paper 2, Section II 20G Number Fields Calculate the class group of the field Q(√ 14 ) . −

Paper 4, Section II 20G Number Fields 3 Suppose that α is a zero of x x + 3 and that K = Q(α) . Show that [K : Q] = 3. − Show that OK , the ring of integers in K, is OK = Z [α] .

[You may quote any general theorem that you wish, provided that you state it clearly. Note that the discriminant of x 3 + px + q is 4 p3 27q2 .] − −

Part II, 2010 List of Questions 2009

60

Paper 1, Section II 20H Number Fields Suppose that K is a number field with ring of integers . OK (i) Suppose that M is a sub-Z-module of K of finite index r and that M is O closed under multiplication. Define the discriminant of M and of , and show that OK disc(M)= r2 disc( ). OK 3 (ii) Describe K when K = Q[X]/(X + 2X + 1). O [You may assume that the polynomial X3 + pX + q has discriminant 4p3 27q2.] − − (iii) Suppose that f, g Z[X] are monic quadratic polynomials with equal discrim- ∈ inant d, and that d / 0, 1 is square-free. Show that Z[X]/(f) is isomorphic to Z[X]/(g). ∈ { } [Hint: Classify quadratic fields in terms of discriminants.]

Paper 2, Section II 20H Number Fields Suppose that K is a number field of degree n = r + 2s, where K has exactly r real embeddings.

(i) Taking for granted the fact that there is a constant CK such that every integral ideal I of has a non-zero element x such that N(x) C N(I), deduce that the class OK | | 6 K group of K is finite. (ii) Compute the class group of Q(√ 21), given that you can take − 4 s n! C = D 1/2, K π nn | K |   where DK is the discriminant of K. (iii) Find all integer solutions of y2 = x3 21. −

Paper 4, Section II 20H Number Fields Suppose that K is a number field of degree n = r + 2s, where K has exactly r real embeddings. Show that the group of units in is a finitely generated abelian group of rank at OK 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 Q(√11).

Part II, 2009 List of Questions 2008 45

1/II/20G Number Fields (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. 1 (b) Put K = Q(ω), where ω = (1 + √ 23), and let K be the ring of integers 2 − O of K. Show that K = Z + Zω. Factorise the ideals [2] and [3] in K into prime ideals. Verify that the equationO of ideals O

[2, ω][3, ω] = [ω] holds. Hence prove that K has class number 3.

2/II/20G Number Fields (a) Factorise the ideals [2], [3] and [5] in the ring of integers of the field OK K = Q(√30). Using Minkowski’s bound

n! 4 s d , nn π | K |   p determine the ideal class group of K. [Hint: it might be helpful to notice that 3 = N (α) for some α K.] 2 K/Q ∈ (b) Find the fundamental unit of K and determine all solutions of the equations x2 30y2 = 5 in integers x, y Z. Prove that there are in fact no solutions of − ± ∈ x2 30y2 = 5 in integers x, y Z. − ∈

4/II/20G Number Fields (a) Explain what is meant by an integral basis of an algebraic number field. Specify such a basis for the quadratic field k = Q(√2).

4 (b) Let K = Q(α) with α = √2, a fourth root of 2. Write an element θ of K as

θ = a + bα + cα2 + dα3 with a, b, c, d Q. By computing the relative traces TK/k(θ) and TK/k(αθ), show that if θ is an algebraic∈ integer of K, then 2a, 2b, 2c and 4d are rational integers. By further computing the relative norm NK/k(θ), show that

a2 + 2c2 4bd and 2ac b2 2d2 − − − are rational integers. Deduce that 1, α, α2, α3 is an integral basis of K.

Part II 2008 2007 43

1/II/20H Number Fields Let K = Q(√ 26). − (a) Show that K = Z[√ 26] and that the discriminant dK is equal to 104. O − − (b) Show that 2 ramifies in by showing that [2] = p2, and that p is not a principal OK 2 2 ideal. Show further that [3] = p3p¯3 with p3 = [3, 1 √ 26]. Deduce that neither p nor p2 is a principal ideal, but p3 = [1 √ 26]. − − 3 3 3 − − (c) Show that 5 splits in by showing that [5] = p p¯ , and that OK 5 5 N (2 + √ 26) = 30. K/Q −

Deduce that p2p3p5 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 2 √104 6.492.] π ≈

2/II/20H Number Fields Let K = Q(√10) and put ε = 3 + √10. (a) Show that 2, 3 and ε + 1 are irreducible elements in . Deduce from the equation OK 6 = 2 3 = (ε + 1)(¯ε + 1) · that is not a principal ideal domain. OK (b) Put p2 = [2, ε + 1] and p3 = [3, ε + 1]. Show that

[2] = p2 , [3] = p p¯ , p p = [ε + 1] , p p¯ = [ε 1] . 2 3 3 2 3 2 3 − Deduce that K has class number 2. (c) Show that ε is the fundamental unit of K. Hence prove that all solutions in integers x, y of the equation x2 10y2 = 6 are given by − x + √10y = εn(ε + ( 1)n) , n = 0, 1, 2,... ± −

Part II 2007 2007 44

4/II/20H Number Fields

Let K be a finite extension of Q and let = K be its ring of integers. We will O O assume that = Z[θ] for some θ . The minimal polynomial of θ will be denoted by g. For a primeO number p let ∈ O

g¯(X) =g ¯ (X)e1 ... g¯ (X)er 1 · · r be the decomposition ofg ¯(X) = g(X)+pZ[X] (Z/pZ)[X] into distinct irreducible monic ∈ factorsg ¯i(X) (Z/pZ)[X]. Let gi(X) Z[X] be a polynomial whose reduction modulo p ∈ ∈ isg ¯i(X). Show that pi = [p, gi(θ)] , i = 1, . . . , r , are the prime ideals of containing p, that these are pairwise different, and O [p] = pe1 ... per . 1 · · r

Part II 2007 2006 41

1/II/20G Number Fields Let α, β, γ denote the zeros of the polynomial x3 nx 1, where n is an integer. The discriminant of the polynomial is defined as − −

∆ = ∆(1, α, α2) = (α β)2(β γ)2(γ α)2. − − − Prove that, if ∆ is square-free, then 1, α, α2 is an integral basis for k = Q(α). By verifying that α(α β)(α γ) = 2nα + 3 − − and further that the field norm of the expression on the left is ∆, or otherwise, show that ∆ = 4n3 27. Hence prove that, when n = 1 and n = 2, an− integral basis for k is 1, α, α2. −

2/II/20G Number Fields Let K = Q(√26) and let ε = 5 + √26. By Dedekind’s theorem, or otherwise, show that the ideal equations

2 = [2, ε + 1]2, 5 = [5, ε + 1][5, ε 1], [ε + 1] = [2, ε + 1][5, ε + 1] − hold in K. Deduce that K has class number 2. Show that ε is the fundamental unit in K. Hence verify that all solutions in integers x, y of the equation x2 26y2 = 10 are given by − ± x + √26y = εn(ε 1) (n = 0, 1, 2,...) . ± ± ± ± 1 [It may be assumed that the Minkowski constant for K is 2 .]

4/II/20G Number Fields Let ζ = e2πi/5 and let K = Q(ζ). Show that the discriminant of K is 125. Hence prove that the ideals in K are all principal. n Verify that (1 ζ )/(1 ζ) is a unit in K for each integer n with 1 6 n 6 4. Deduce that 5/(1 ζ−)4 is a unit− in K. Hence show that the ideal [1 ζ] is prime and totally ramified in−K. Indicate briefly why there are no other ramified prime− ideals in K. [It can be assumed that ζ, ζ2, ζ3, ζ4 is an integral basis for K and that the Minkowski constant for K is 3/(2π2).]

Part II 2006 2005 39

1/II/20G Number Fields Let K = Q(√2, √p) where p is a prime with p 3 (mod 4). By computing the ≡ relative traces TrK/k(θ) where k runs through the three quadratic subfields of K, show that the algebraic integers θ in K have the form

1 1 √ θ = 2 (a + b√p) + 2 (c + d√p) 2 , where a, b, c, d are rational integers. By further computing the relative norm NK/k(θ) where k = Q(√2), show that 4 divides

a2 + pb2 2 c2 + pd2 and 2 ab 2cd . − −

Deduce that a and b are even and c d (mod 2). Hence verify that an integral basis for K is ≡ √ 1 √ 1, 2, √p, 2 1 + √p 2.

2/II/20G Number Fields Show that ε = (3 + √7)/(3 √7) is a unit in k = Q(√7). Show further that 2 is the square of the principal ideal in−k generated by 3 + √7. 1 Assuming that the Minkowski constant for k is 2 , deduce that k has class number 1. Assuming further that ε is the fundamental unit in k, show that the complete solution in integers x, y of the equation x2 7y2 = 2 is given by − x + √7y = εn(3 + √7) (n = 0, 1, 2,...). ± ± ± Calculate the particular solution in positive integers x, y when n = 1.

4/II/20G Number Fields State 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 Q(√d) with d a square- free 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 Q(ζ), where ζ = e2πi/q with q an 3 odd prime, and also in the cubic field Q(√2). [The cases d 2, 3 (mod 4) and d 1 (mod 4) for the quadratic field should be carefully ≡ ≡ q 2 distinguished. It can be assumed that Q(ζ) has a basis 1, ζ, . . . , ζ − and discriminant (q 1)/2 q 1 3 3 3 2 ( 1) − q − , and that Q(√2) has a basis 1, √2, (√2) and discriminant 108.] − −

Part II 2005