An Introduction to Galois Theory Solutions to the exercises

1-2. (a) Let u, v ∈ S and suppose that uv = 0; then u = 0 or v = 0 since u, v ∈ R and R is an integral domain. Consider the unit homomorphisms η : Z −→ R and η′ : Z −→ S. Then for n ∈ Z, η′(n) = η(n), so ker η′ = ker η and therefore charS = charR. (b) Q is a field and Z ⊆ Q is a subring which is not a field. 1-3. (a) For any subring R ⊆ C, R is an integral domain with characteristic subring Z and charR = 0. (b) The characteristic subring of A[X] is the same as that of A and charA[X] = charA. A[X] is an integral domain if and only if A is an integral domain. (c) If we identify A with the subring of scalar matrices in Matn(A), then the characteristic subring of Matn(A) is the same as that of A and charMatn(A) = charA. If n > 1 then Matn(A) is not commutative, in any case it always has zero-divisors since any singular matrix is a zero-divisor.