MP473 Examination, November 2000
Time: 3 hours
Answer all questions
1. (a) Explain what is meant by the statements: (i) Kis an algebraic
number field of degree n, (ii) ω1, . . . , ωnis an integral basis for K,
(iii) UKis the unit group of OK, (iv) DK, the discriminant of K,
(v) TK(α), the trace of α(vi) NK(α), the norm of α.
(b) Write down integral bases and discriminants for Q(√d), when
d=−5 and d= 29. Also describe explicitly the unit group UKin
each case.
(c) Give an example in K=Q(i), where αand βbelong to OK,
NK(α) divides NK(β) in Z, but αdoes not divide βin OK.
2. Let f(x) = x4+ 2x2−2∈Q[x]..
(a) Prove that f(x) = x4+ 2x2−2 is irreducible in Q[x].
(b) Let θ4+ 2θ2−2 = 0 and let K=Q(θ).
(i) Prove that 2/θ2∈OKand that 2/θ4∈UK. Also prove that
2/θ4= 3 + θ2.
(ii) Verify that 3 = (1+ θ2)2and explain what this tells us about
DK.
(iii) Given that ∆K(1, θ, θ2, θ3) = −2932, explain why DK=−2932.
3. (a) If Iis an ideal of OK, define N(I), the norm of I.
(b) If Aand Bare ideals of OKand A+B={a+b|a∈A, b ∈B},
prove that A+Bis an ideal and that A+B= gcd (A, B ).
(c) If K=Q(√−5) and A= (2,1 + √−5), prove directly from the
definition, without appealing to the Kummer–Dedekind theorem,
that NK(A) = 2.
(d) Let Ibe an ideal of OKand let α∈Isatisfy
N(I) = |NK(α)|.
Prove that I= (α).
(e) Let Kbe a real quadratic field and suppose that the fundamental
unit ηof Ksatisfies NK(η) = 1. Let Ibe an ideal of OKwith the
property that
I2= (α),
where NK(α)<0. Prove that Iis not a principal ideal.
4. Let K=Q(√−17).
(a) Show that 2 and 3 are the only primes which must be examined
in order to determine the ideal class group IK.
(b) Let ω=√−17. Use the Kummer–Dedekind theorem to factorise
(2) and (3): (2) = Q2,(3) = P R, where
Q= (2,1 + ω), P= (3,1 + ω), R= (3,−1 + ω).
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