Math 250a: Higher Algebra
Problem Set #2 (6 October 2004): Galois theory II
1. Prove that Zis the only subring of Qthat is finitely generated as a module over Z, and
conclude that Zis integrally closed in Q.
2. (Proof of the result mentioned at the end of the notes on integral closure) Let Abe a
subring of some field F, and assume that Ais integrally closed in F. Let ube an
element of some field K/F which is algebraic over Fand integral over A. Prove that
the minimal monic polynomial of uis contained in A[X]. (Hint: Factor this polynomial
over its splitting field.)
3. (Fermat’s last theorem in F[X]) Suppose A, B, C ∈F[X] are polynomials satisfying
A+B+C= 0, and let W=AB 0−A0B. Show that if ris a root of A,B, or Cof
multiplicity min some extension field K/F then ris a root of Wof multiplicity at
least m−1.
Use this to prove that if Fis a field of characteristic zero then for each integer n≥3
the Fermat equation xn+yn=znhas no solution in relatively prime polynomials
x, y, z ∈F[X] of positive degree. Does this remain true in characteristic p > 0? If not,
what additional condition must be imposed on x, y, z , n?
[The method can be generalized to xn+yn+zn=tn, etc., but imperfectly: already with four
terms it is still not known what is the largest nfor which xn+yn+zn=tnhas a nontrivial
solution in C[X]. Can you prove n < 8? Can you find an example with n > 4?]
4. (Problem 2 of Jacobson 4.4) Let Fb e a field of characteristic p. Prove that every irre-
ducible polynomial f∈F[X] can be written as g(Xpe
) for some irreducible separable
polynomial g∈F[X] and some nonnegative integer e. Use this to show that every root
of f(in a splitting field of f) has the same multiplicity pe.
5. (Problem 4 of Jacobson 4.5, generalized) Let E=C(t), the field of rational functions
over Cin a transcendental t. Fix a positive integer nand a primitive n-th root of
unity ω∈C[that is, a generator of the group of n-th roots of unity; for example,
ω=e2πi/n)]. Let σ, τ be the the following automorphisms of E:
(σf )(t) := f(ωt); (τ f )(t) := f(1/t).
Show that σn=τ2= (στ)2= id. Determine the structure of the group Ggenerated
by σand τ, and prove that the subfield Fof Efixed by Gis C(u) where u=tn+t−n.
6. (Problem 3 of Jacobson 4.4, generalized) Let kbe a finite field qelements, and Fa field
contianing k. A polynomial f∈F[X] is called a q-polynomial if it is of the form
Pm
i=0 aiXqi
for some ai∈F. Prove that a polynomial f∈F[X] of positive degree is a
q-polynomial if and only if its roots form a k-vector subspace of Fand each root has
the same multiplicity which is of the form qefor some nonnegative integer e.
[Note that the polynomial Xq
−Xwe used to construct kis a special case, and is a q0-polynomial
for every q0such that qis a power of q0.]
7. (Problem 3 of Jacobson 4.5) Let Fbe a field of characteristic p, and aan element of F
not in {bp−b|b∈F}. Prove that the polynomial Xp−X−ais irreducible over F,
and determine its Galois group. Can you obtain a generalization with preplaced by a
prime power q?
Problem set is due in class Wednesday the 13th.