Math 503 Problem Set #12 Due the week of April 9, 2007, in lab.
Read Artin, Chapter 13, sections 6-9; and Chapter 14, section 1.
Part A:
From Artin, do these problems (given at the end of Chapter 13): Section 13.5: 1; 13.6: 7,
10; 13.8: 1; miscellaneous problems: 2.
From Artin, do these problems (given at the end of Chapter 14): Section 14.1: 3, 6(a,c).
Part B:
1. Let pbe a prime number, and let r, s be positive integers. Show that rdivides sif and
only if pr−1 divides ps−1.
2. Let Kbe a field, and f(x)∈K[x]. Assume that Khas characteristic 0. Let n≥1.
a) Let Lbe a finite field extension of K, and let α∈L. Show that αis a root of f
with multiplicity exactly nif and only if 0 = f(α) = f0(α) = ··· =f(n−1) (α)6=f(n)(α).
b) Show that fhas a root (in some extension of K) of multiplicity at least nif and
only if (f(x), f 0(x), . . . , f(n−1) (x)) is a proper ideal of K[x].
c) What if instead Khas non-zero characteristic?
3. Let pbe a prime number and let E=Q[ζp], where ζpis a primitive pth root of unity.
Let F=Q.
a) Find the degree and the Galois group Gof the field extension F⊂E.
b) Determine if the extension is separable and normal.
c) Find the fixed field of Gin E.
d) Is the extension Galois?
Part C:
From Artin, do these problems (at the end of Chapter 13):
Section 13.6: 15; miscellaneous problems: 3 [Hint: Consider the quadratic factors of this
polynomial over the splitting field, and then apply miscellaneous problem 2(a) with α= 2,
β= 3].
From Artin, do these problems (at the end of Chapter 14):
Section 14.1: 5, 14.
Also do the following problem:
Let pbe a prime number, and let Fbe a field that contains a primitive pth root of
unity ζp(i.e. ζp
p= 1 but ζn
p6= 1 for 0 < n < p).
a) Show that Fdoes not have characteristic p.
b) Let a∈Fbe an element that is not a pth power in F, and let E=F[p
√a]. Repeat
parts (a)-(d) of problem B3 for this field extension F⊂E.