Math 603 Problem Set #9 Due Mon., April 11, 2005
1. Let F=Z/pZ, let L=F(x, y), and let K=F(xp, y p). Show that Lis a finite field
extension of K, but that there are infinitely many fields between Kand L. Is L=K[α]
for some α∈L? Is Lseparable over K?
2. Let ζn=e2πi/n ∈Cand let Φn(x) be the minimal polynomial of ζnover Q.
a) Find the roots of Φn(x). Show that deg Φn(x) = φ(n), where
φ(n) = #{m∈Z|1≤m≤n, (m, n) = 1}.
b) Show that Y
d|n
d>0
Φd(x) = xn−1, and deduce that X
d|n
d>0
φ(d) = n.
c) Let Kn=Q(ζn). What is [Kn:Q]? [Hint: Observe Kn≈Q[x]/(Φn(x)).]
d) Show that Knis Galois over Q.
3. Let Kbe a field, and let x1, . . . , xnbe transcendentals (variables) over K. For i=
1, . . . , n let sibe the ith elementary symmetric polynomial in x1, . . . , xn. (Thus s1, . . . , sn
are algebraically independent.)
a) Show that K(x1, . . . , xn) is the splitting field over K(s1, . . . , sn) of the polynomial
Zn−s1Zn−1+s2Zn−2− · ·· + (−1)nsn.
b) Show that the extension K(s1, . . . , sn)⊂K(x1, . . . , xn) is Galois.
c) Show that the Galois group Gis the symmetric group Sn. [Hint: Show that Sn⊂G
and that #G= [K(x1, . . . , xn) : K(s1, . . . , sn)] ≤n!.]
4. Let Lbe a normal field extension of K, and let K0be the maximal purely inseparable
extension of Kcontained in L. View Las contained in a fixed algebraic closure ¯
Kof K.
a) Let β∈L, and let β1, . . . , βn∈¯
Kbe the distinct images of βunder the K-
embeddings L ,→¯
K(listing each image only once, regardless of multiplicity). Let f(x) =
Q(x−βi). Show that f(x)∈K0[x]. [Hint: Show f(x)∈L[x], and then show that the
coefficients of fare mapped to themselves under each K-embedding L ,→¯
K.]
b) In part (a), show that f(x) is the minimal polynomial of βover K0. [Hint: Show
that every βiis a root of the minimal polynomial.]
c) Conclude that Lis separable over K0.
5. Let K=Fp(t) and let L=K[2p
√t], where pis an odd prime number.
a) Find the maximal separable extension K0of Kin L, and the maximal purely
inseparable extension K0of Kin L.
b) Show explicitly in this example that Lis the compositum of K0and K0, by ex-
pressing 2p
√tas a combination of elements from K0and K0.
c) What if instead p= 2?
The following problems are optional.
6. Prove “Fermat’s Last Theorem” for the ring C[t] (rather than for Z, as usual). That
is, show that if n > 2 then there is no choice of relatively prime non-constant polynomials
p(t), q(t), r(t)∈C[t] such that pn+qn=rn. Do this in steps, as follows:
