Math 402/502: Sample Homework Solutions
(Ch. 12 # 20) Describe the elements of M2 (Z) that have multiplicative inverses.
The elements in M2 (Z) with inverses are invertible 2 × 2matrices
Z,
with entries from
d −b
a b
1
, then
whose inverses also have entries from Z. Since the inverse of
is ad−bc
−c a
c d
the determinant ad − bc must divide each of a, b, c and d. This is certainly satisfied if
ad − bc = ±1. To see that these are the only possible determinants, write a = t1 (ad − bc),
b = t2 (ad − bc), c = t3 (ad − bc), and d = t4 (ad − bc). Then ad − bc = (ad − bc)2 (t1 t4 − t2 t3 ),
so since ad − bc 6= 0, we have 1 = (ad − bc)(t1 t4 − t2 t3 ). Then since ad − bc is an integer, we
must have ad − bc = ±1. So the units of M2 (Z) are the matrices with determinant 1 or −1.
(Ch. 13 # 31) Let F be a field of order 2n . Prove that char F = 2.
Since F is an integral domain, its characteristic must be 0 or prime. But since F is finite,
every element has finite additive order, so the characteristic is nonzero, and hence prime.
Since F has a unity, the characteristic p is the additive order of 1. Hence F , viewed as a
group, has a cyclic subgroup of order p. Then by Lagrange’s Theorem, p | |F |, so we must
have p = 2.