Math 503 Problem Set #10 Due the week of March 26, 2007, in lab.
Read Artin, Chapter 13, sections 1-3.
Part A:
From Artin, do these problems (given at the end of Chapter 12): Miscellaneous problems
8, 9.
From Artin, do these problems (given at the end of Chapter 13): Section 13.1: 1, 4; 13.2:
1; 13.3: 1, 2.
Part B:
1. a) If Ris a commutative ring, find an isomorphism R⊗ZZ[√2] →
∼R[x]/(x2−2).
b) Determine whether Z[√3] ⊗ZZ[√2] and Z[√2] ⊗ZZ[√2] are integral domains.
c) Simplify each of the following Z-modules (up to isomorphism):
Z/10 ⊗ZZ,Z/10 ⊗ZZ/6, Z/10 ⊗ZQ,Q⊗ZQ,Z/10 ⊗Z3.
2. Let Rbe a commutative ring, let φ:N1→N2be a homomorphism of R-modules, and
let Mbe an R-module.
a) Show that there is an induced homomorphism of R-modules φ∗:M⊗N1→M⊗N2
defined by φ∗(m⊗n) = m⊗φ(n).
b) Show that if φis surjective then so is φ∗.
c) Show that if Mis a free R-module (e.g. if Ris a field), then if φis injective then so
is φ∗. But show by example that if Mis arbitrary, then it is possible for φto be injective
and φ∗not to be injective. [Compare PS #9, problem B2.]
3. For each of the following field extensions Fof Q, find the degree of Fover Qand find
the group Aut(F) of automorphisms of F.
Q,Q[√5], Q[ζ5] (where ζ5is a primitive fifth root of unity), Q[4
√2], Q[5
√2]
Part C:
From Artin, do these problems (at the end of Chapter 13):
Section 13.1: 2, 3; 13.2: 3.
Also do the following problem:
Let K=Q[√2] and L=Q[p2 + √2].
a) Find the multiplicative inverse of p2 + √2inL(as a polynomial in p2 + √2).
b) Show K⊂L. What is [K:Q]? [L:K]? [L:Q]?
c) Let φbe an automorphism of L. What can you say about the restriction φ|Q?
d) Let φbe an automorphism of L. What can you say about the restriction φ|K?
e) Find an element of order 4 in Aut(L). What is the group Aut(L) abstractly?
