Math 250: Higher Algebra
Problem Set #1 (24 September 2004): Galois theory I
1. (Another construction of the trace and norm) Let K/F be a finite field extension with
[K:F] = n. For each a∈K, we may consider the map Ma:K→K,x7→ ax as a linear
operator on Kconsidered as a vector space over F.
i) Check that a7→ Mais a homomorphism from Kto EndF(K), the algebra of
F-linear operators on K.
The trace and norm of a(relative to the extension K/F ) are the trace and determinant
of Ma. These are denoted TrK/F(a) and NK/F (a), or simply Tr(a) and N(a) if K/F is
understood.
ii) Check that Tr is an F-linear map from Kto F, and that the norm is multi-
plicative: N(ab) = N(a)N(b) for all a, b ∈K. If F=Rand K=C, what are
the trace and norm of a=x+iy? What are the eigenvalues of Ma?
It is a fundamental result in linear algebra that a linear operator Ton a finite-dimensional
vector space satisfies P(T) = 0 where P(λ) = det(λI −T) is the characteristic polynomial
of T. This gives an explicit construction of a monic polynomial (NB not always the minimal
such polynomial!) satisfied by an element of K.
iii) Suppose K=F(u) where uis a root of the irreducible polynomial P(X) =
Xn+Pn−1
j=0 ajXjin F[X]. Determine the matrix of Murelative to the F-basis
{1, u, u2, . . . , un−1}of K, and check directly that Pis the characteristic polyno-
mial of this matrix.
2. (An application of part (iii) of the last problem; cf. problem 2 of Jacobson 4.1) Let F=Q,
K=Q(√2,√3 ), and a=√2 + √3∈K. Determine n= [K:F]. Prove that K=F(a).
(Hint: what can [K:F(a)] be?) Choose a basis for Kas a F-vector space, and determine
the matrix of Marelative to this basis. Use this to compute the minimal p olynomial of a
over F. Check directly that this polynomial vanishes at a.
3. (Problem 7 of Jacobson 4.1) A field extension L/F is said to be algebraic if every element
of Lis algebraic over F. Suppose L/F is algebraic and K⊆Lis an F-subalgebra, i.e., a
subring containing F(equivalently, an F-vector subspace containing Fand closed under
multiplication). Prove that Kis a field.
4. (Problem 8 of Jacobson 4.1) Let L/F be the transcendental extension F(u). Suppose that
Kis a subfield of Lproperly containing F. Prove that uis algebraic over K.
5. (Problem 2 of Jacobson 4.3) Construct a splitting field Kof x5−2 over Q, and determine
[K:Q]. (You may assume the irreducibility of x5−2 over Q, and of the polynomial
(x5−1)/(x−1) over Q(5
√2); we shall learn later how to prove such results.)
6. (Problem 4 of Jacobson 4.3) Let L/F be a splitting field over Fof some polynomial f(X),
and let Kbe any subfield of Lcontaining F. Suppose ι:K→Lis a homomorphism
whose restriction to Fis the identity. Prove that ιcan be extended to an isomorphism
of L.
7. (Problem 4 of Jacobson 4.4) Let Fbe a field of characteristic pthat is not perfect. Thus
there are elements of Fnot contained in Fp; let abe any such element. Prove that the
polynomial Xpe
−ais irreducible for every nonnegative integer e.
8∗. (Curious behavior of an inseparable extension) Let kbe a field, K=k(X, Y ) the field
of rational functions in two variables, and F=k(Xp, Y p) for some prime p. Show that
[K:F] = p2. Now assume kis an infinite field of characteristic p. (For instance we
may have k=k0(T) with k0=Z/pZ.) Prove that there are infinitely many intermediate
fields Ebetween Kand F(necessarily with [K:E] = [E:F] = p).
We shall see that this cannot happen if K/F is a separable extension of finite degree.
Problem set is due in class Monday, October 4th.
