Algebraic Number Theory
Math 514B Spring 2008
Problem Set 1
Due: Tuesday, Jan. 29th
1. a) Let K⊂Lbe a fininte extension of number fields, and let ˜
Lbe the Galois
closure of Lover K. Show that SplL/K =Spl˜
L/K . (Hint: Let pbe a prime
of K, and let Pbe a prime of Lover p. Let Kpbe the p-adic completion of
K, and let ¯
Kpbe its algebraic closure. Pick an embedding α:L ,→¯
Kpas K-
algebras. Show that the condition that a prime pis in either Spl is equivalent
to saying that every conjugate of Lhas the property that its image under α
is contained in Kp.)
b) Consider the result that if L, E are finite Galois extensions of a number field
K, then
SplL/K ⊂SplE/K ⇐⇒ E⊂L.
Does this assertion remain true if Lis permitted not to be Galois? If Eis
permitted not to be Galois? If both are permitted not to be Galois? In each
case, either give a proof or a counterexample.
2. Fix a rational prime p.
a) Show that for each positive divisor nof p−1 (i.e. p≡1 mod n) there exists a
cyclic Galois extension K/Qof degree nsuch that pis the only finite ramified
prime.
b) Explicitly write down a cubic polynomial that generates a normal cubic ex-
tension Kof Qfor which 7 is the only finite ramified prime.
c) Explicitly write down a cubic polynomial that generates a normal cubic ex-
tension Kof Qfor which 3 is the only finite ramified prime.
3. Fix an integer n > 1 along with a rational prime pnot dividing n. Denote R=Z[ζn]
as the integral closure of Zin K=Q(ζn), and pas a prime in Rlying over p.
a) Show that R/pR∼
=Fp[ζn], and that pis unramified in R.
b) Show that ζn∈Fpif and only if p≡1 mod n. Conclude that psplits
completely in Rif and only if p≡1 mod n.