Math 232
Problem set #2
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(1) Let K = Q( 3 5). You may assume that OK = Z[ 3 5]. For every rational
prime p, show how pOK factors into prime ideals in OK , and for each prime
ideal factor p of pOK find |OK /p|.
(2) Suppose K ⊂ L ⊂ F are number fields, and F/K is Galois. Fix a prime p of
K, a prime pL of L above p, and a prime pF of F above pL . Write GpF (F/K)
for the decomposition group of pF in Gal(F/K), IpF (F/K) for the inertia
group of pF in Gal(F/K), etc.
(a) Show that GpF (F/L) = GpF (F/K)∩Gal(F/L) and IpF (F/L) = IpF (F/K)∩
Gal(F/L).
(b) If L/K is Galois, show that GpL (L/K) (resp. IpL (L/K)) is the image of
GpF (F/K) (resp. IpF (F/K)) in Gal(L/K).
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(3) Let K = Q( −5). Let p be the ideal generated by 2 and 1 + −5.
(a) Show that p is a prime ideal of OK .
(b) Give generators for the fractional ideal p−1 .
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(4) Let L = Q( −3, 3 5), the splitting field of x3 − 5. If p is a rational prime,
describe as best you can the splitting of p in L (for example, the number of
primes of L above p, and the decomposition groups and inertia groups of those
primes). You might find it helpful to use problems (1) and (2), and problem
(6) from problem set #1.