Math 250a: Higher Algebra
Problem Set #8 (29 November 2004):
Tensor products etc.; Chevalley’s Theorem
Some overdue basics on tensor products of algebras:
1. i) Let Abe any algebra over a field k. Prove that A⊗kMn(k) is isomorphic
as a k-algebra with Mn(A), for each positive integer n.
ii) Prove that the tensor product Mm(k)⊗kMn(k) is isomorphic as a k-algebra
with Mmn(k) for all positive integers m, n (and an arbitrary commutative
field k).
An instructive example of a division algebra in positive characteristic:
Let kbe a finite field, and Athe noncommuting k-algebra generated by two
indeterminates x, y satisfying xy −yx = 1.
2. i) Prove that Ahas no zero divisors, and that its center consists of the poly-
nomials in X:= xpand Y:= yp. (Hint for the second part: compare xya
with yax, and likewise xbywith yxb.)
ii) Let R=k[X, Y ] be that center, and Kits fraction field (the field of rational
functions in two variables with coefficients in k). Let D=A⊗RK. Show
that Dis a skew field with center K.
iii) We know that Dmust have a separable decomposition field. Find an
inseparable field extension K0/K of degree psuch that D⊗KK0∼
=Mp(K0).
Can you find, at least for p= 2 or p= 3, an explicit isomorphism by finding
endomorphisms x, y of K0psatisfying xp=X,yp=Y, and xy −yx = 1?
Finally, we obtain Chevalley’s theorem on solutions of polynomial equations in
many variables over a finite field, and deduce the triviality of Br(k).
Fix a finite field k=Fq, and let pbe its characteristic, so qis some power of p.
3. i) Prove that Px∈kxm= 0 for all nonnegative integers m < q −1. What is
Px∈kxmfor an arbitrary nonnegative m∈Z?
It follows that Px∈kP(x) = 0 for any polynomial P∈k[X] of degree less than
q−1. We next extend this to polynomials in several variables X1, . . . , Xn. The
“degree” of a nonzero monomial cQn
i=1 Xmi
iis Pimi; the degree of a sum of
distinct monomials is the largest of those monomials’ degrees. This defines the
degree on k[X1, . . . , Xn]. (Note that this degree is invariant under an invertible
linear change of variables; thus we may speak of a polynomial of degree mon an
n-dimensional vector space over kwithout specifying which coordinates on that
space we use.)
ii) Prove that
X
(x1,...,xn)∈kn
P(x1, . . . , xn) = 0
for every polynomial P∈k[X1, . . . , Xn] of degree less than n(q−1).
