Download MATH 225A PROBLEMS OCTOBER 2, 2012 (1)

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.)