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MATH 225A PROBLEMS OCTOBER 2, 2012 (1) Which√of the√following √ numbers are algebraic integers: (i) 12 15( 39 7 + 7 39) √ (ii) 1+i 2 (iii) √ √ 1+ 3 10+ 3 100 3 √ (2) (i) Let d be a squarefree integer. Find the ring of integers of Q( d). (ii) √ Let d be a squarefree integer with the property that d ≡ 1 (4). Show that Z( d) is not a PID. (3) Let L/K be a finite, separable extension of fields (not necessarily of characteristic 0). (i) Show that T rL/K : L × L → K; (x, y) 7→ T rL/K (xy) is a non-degenerate, symmetric, K-bilinear form on L. (ii) Show that the map T rL/K : L → K; x 7→ T rL/K (x) is surjective. (Hint for both (i) and (ii): Artin’s theorem on linear independence of characters.) (4) Suppose that K is a number field, and let x ∈ OK . Show that x is a unit in OK if and only if NK/Q (x) = ±1. (Hint: You may find it helpful to view K as a Q-vector space and then consider the characteristic equation of the “multiplication by x” map on K.) P κi (5) Let ζ nP = 1 and assume that α = m1 ( m i=1 ζ ) is an algebraic integer. Show that κi either m = 0 or ζ κ1 = ζ κ2 = · · · = ζ κm . i=1 ζ (Hint: Set K = Q(ζ). First consider NK/Q (α), and see what this tells you about α. Then think geometrically. ) √ (6) Find the ring of integers, and calculate the discriminant of Q( 3 5). √ √ (7) Find the ring of integers, and calculate the discriminant of Q( 2, i). (8) Let θ be a root of the polynomial T 3 − 2T + 2. Calculate the ring of integers of Q(θ). (9) Let K be a number field with [K : Q] = n, and let 2t of the n embeddings of K into C have complex image. By considering the action of complex conjugation on ∆(K/Q), show that the sign of d(K/Q) is equal to (−1)t . (10) Let K be a number field with [K : Q] = 3. By considering the action of various Galois embeddings on ∆(K/Q), show that d(K/Q) is a square if and only if K/Q is Galois. (11) Let f (T ) ∈ Z[T ] be a monic, irreducible polynomial. Let x be a root of f , and let K = Q(x). By expressing 1/f (T ) in partial fractions, show that TrK/Q (xi /f 0 (x)) = 0 =1 1 0≤i<n−1 i=n−1 2 MATH 225A PROBLEMS OCTOBER 2, 2012 Now suppose that OK = Z[x]. Let D−1 denote the image of the dual of OK under the composite homomorphism HomZ (OK , Z) ,→ HomQ (K, Q) ' K, where the isomorphism is given by the trace. Show that D−1 is a fractional ideal and that N (D) = |d(K/Q)|. (12) Let p be an odd prime, and let ζp be a primitive pth root of unity. Set Γ = Gal(Q(ζp )/Q), and let χ : Γ → C∗ be a character of order n > 1 (i.e. χ is a group homomorphism, and n is the least integer such that χn is the trivial homomorphism). We define the Gauss sum τ (χ, ζp ) by X τ (χ, ζp ) = χ(γ)ζpγ . γ∈Γ (i) Show that, for γ ∈ Γ, we have τ (χ, ζpγ ) = χ(γ −1 )τ (χ, ζp ). (ii) Show that τ (χ, ζp )τ (χ, ζp ) = p. (Here z denotes the complex conjugate of z.) (iii) Let χ be the unique character of Γ of order 2. From (i) and (ii), deduce that s −1 τ (χ, ζp ) = ± p. p √ (13) Describe the factorisation of the ideals generated by 2, 3, 5 in Q( 3 6). (14) Let θ satisfy θ3 − θ − 1 = 0. Describe the factorisation of the ideals generated by 2, 3, 5, 23 in Q(θ). (15) Let p be a prime and a be an integer, and let (a/p) denote the Legendre symbol. Taking a to be an integer modulo p, verify that the map from F∗p to {±1} given by a 7→ (a/p) is a homomorphism. Let p be an odd prime and let z be a generator of the multiplicative group F∗p . Show that z (p−1)/2 = −1 and hence deduce Euler’s criterion: For any d prime to p, d(p−1)/2 ≡ (d/p) mod p. (16) Let p, q be distinct odd primes and let w denote a primitive pth root of unity in an extension of Fq . For any a ∈ F∗p , define the Gauss sum (in an extension of Fq as X x τ (a) = wax . p x∈F∗ p Prove: (i) τ (a) = (a/p)τ (1), (ii) τ (1)q = τ (q), (iii) τ (1)2 = (−1)(p−1)/2 p. (17) For any odd n, put ε(n) ≡ (n − 1)/2 mod 4. Use (2) above to show that τ (1)q−1 = (q/p). By evaluating τ (1)q−1 = [τ (1)2 ](q−1)/2 in two ways, prove the law of quadratic reciprocity, viz.: p q = (−1)ε(p)ε(q) . q p MATH 225A PROBLEMS OCTOBER 2, 2012 3 (18) For any odd n, put ω(n) ≡ (n2 − 1)/2 mod 8. Let α be a primitive 8th root of unity in an extension of Fp , and put β = α + α−1 . Show that β 2 = 2. Using the Frobenius endomorphism x 7→ xp and Euler’s criterion to evaluate β p−1 in two ways, prove that (2/p) = (−1)ω(p) . √ √ (19) Find the class numbers of Q( −2) and Q( −6). Hence find all the integral solutions to (i) x3 = y 2 + 2 (ii) x3 = y 2 + 54 (20) For an integer n, let ζn denote a primitive nth root of unity. (i) If n is not a prime power, show that 1 − ζn is a unit of Q(ζn ). (ii) Let p be a prime, and let (m, p) = 1. Show that (1 − ζp )/(1 − ζpm ) is a unit of Q(ζp ). √ (21) Calculate the class number of K := Q( 3 2). Find a unit of infinite order in K. √ (Recall that we have shown that Z( 3 2) is the ring of integers of K.)