Math 154. Homework 7
1. A discrete valuation ring (dvr) is a Dedekind domain A with a unique maximal ideal.
(i) Via Z/7Z Z(7) /7Z(7) , nd n Z so n mod 7Z goes to 2/3 mod 7Z(7) . Express n 2/3 as 7x with
x Z(7) . For prime p, prove Z consists of q Q with n
Math 154. Discriminant of composite fields
Let F be a number eld which is a compositum (over Q) of two subelds K and K . We assume that
[F : Q] = [K : Q][K : Q], or equivalently that [F : K ] = [K : Q] (or equivalently [F : K ] = [K : Q]). Under
this assu
Math 154. Integer ring of prime-power cyclotomic field
r
Let p > 0 be a prime number, and consider the splitting eld K of X p 1 over Q for a xed r 1. We
saw in class that K = Q(pr ) for any primitive pr th root of unity pr , [K : Q] = (pr ) = pr1 (p 1), a
Math 154. Class groups for imaginary quadratic fields
In general it is a very dicult problem to determine the class number of a number eld, let alone the
structure of its class group. However, in the special case of imaginary quadratic elds there is a ver
Math 154. Homework 4
1. (i) Prove that if n 1 is odd and n is a primitive nth root of unity then n is a primitive 2nth root of
unity. In particular, Q(n ) = Q(2n ) when n is odd. The rest of this exercise shows via Galois theory that
this is the only case
Math 154. Dedekinds factorization criterion
The aim of this handout is to give a proof of Dedekinds criterion for computing the prime factorization
of pOK for a prime number p > 0 and a number eld K. The initial setup is to consider OK that is
primitive f
Math 154. Homework 1
0. Read the handout on norm and trace, and then do the
following calculations.
(i) If K = k( a) (a k a nonsquare) then for = x + y a show TrK/k () = 2x and NK/k () = x2 ay 2 .
(ii) For the biquadratic eld K = Q( 2, 3) and = x + y 2
Math 154. Homework 3
0. Read the proof of Proposition 2 in 2.1 of the text (integrality of ring extensions is transitive).
(i) Deduce that if K /K is an extension of number elds then not only is OK integral over OK (even
over Z!) but it is the integral cl
Math 154. Homework 2
1. (i) By using unique factorization and your knowledge of units in Z[i], prove that if p is a positive prime
in and p = x2 + y 2 then (x, y) is unique up to rearrangement and signs. Likewise, using arithmetic in
Z
Z[ 2], prove that i
Math 154. Homework 5
1. (i) Read 1.3 of the text, and using Corollary 2 there show that (n) > n for all n > 6.
(ii) For a number eld K, give a (crude) upper bound in terms of [K : Q] on n such that K contains a
primitive nth root of unity.
(iii) Explain w
Math 154. Unique factorization in Dedekind domains
1. Background
In class we dened a Dedekind domain to be a noetherian domain A that is integrally closed in its fraction
eld F such that (i) all nonzero prime ideals p of A are maximal, and (ii) there exis
Math 154. Norm and trace
An interesting application of Galois theory is to help us understand properties of two special constructions
associated to eld extensions, the norm and trace. If L/k is a nite extension, we dene the norm and trace
maps
NL/k : L k,
Math 154. Modules over a PID
In this handout we give prove a special case of the structure theorem for modules over a PID,
sucient for our purposes in this course (though the general structure theorem is certainly needed
for more advanced applications in
Math 248A. Some vertical factorizations
Let us recall the general result on vertical factorization as discussed in class. We let A be a Dedekind
domain with fraction eld F , and let F /F be a nite separable extension such that the integral closure A of
A
Math 154. Homework 6
1. Let A be a Dedekind domain, F its fractional eld, and IA denote the group of fractional ideals of A.
(i) A fractional ideal ideal I of A is principal if I = Ax for some x F . Prove that the set PA of principal
fractional ideals of
Math 154. A non-primitive ring of integers
The aim of this handout is to explain Dedekinds example of a cubic eld K for which OK does not have
the form Z[] for any integral primitive element of K . In fact, well even see that whatever we choose,
the index