Department of Mathematical Sciences
Instructor: Daiva Pucinskaite
Algebraic Number Theory
November 4, 2016
Name:
Homework Assignment 8
1. Consider L = Q[α], where m(x) = x3 − 5x + 5 is the minimal polynomial of α, and
OL = Z[α].
(a) For which prime numbers p, the ideal (p) ramifies in OL .
(b) For all prime numbers p ≤ 7 compute explicitly the decomposition of (p) = pOL as a
product of prime ideals.
(c) For which prime numbers p ≤ 7 the ideal (p) splits in OL , and (p) is inert.
2. Let α ∈ OL , f (X) = a0 X n + a1 X n−1 + · · · + an , g(X) = b0 X n + b1 X n−1 + · · · + bn ∈ Z[X],
and m ∈ N. Consider f (X) = a0 X n + a1 X n−1 + · · · + an and g(X) = b0 X n + b1 X n−1 + · · · + bn
as the polynomials in Z/(m)[X]. Show: If
f (X) = g(X), then (m, f (α)) = (m, g(α)).
(Here (a,
b) is the ideal of OL generated by a and b, i.e. (a, b) = aOL + bOL .
)
3. Let [L : Q] = n, and m ∈ N. Show:
Nm((m)) = (m)n .
4. Let a be an ideal of OL , and N(a) = |OL /a| be the numerical norm of a.
(We have N(a) ∈ a.)
(a) Let I be an ideal of OL . Show that the set {a is an ideal of OL | I ⊆ a} is finite.
(b) Show: For any m ∈ N the set Sm := {a is an ideal of OL | N(a) = m} is finite.
(c) Let [L : Q] = n, and d ∈ N. Show: If d ∈ a, then N(a) divides dn .