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<m2, − then n2 + n + m is prime. (b) Show that the ideal class group of Q(√ 163) is trivial. − 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.