16.1.14. Let R = M2×2 (R) be the ring of two-by-two matrices with real entries. [0 b [a b] Let S = { | a, b = R} and T = { 0 a 00 [o of] b € R} . (a) Are T and S subrings of R? (b) Is T an ideal of S? Is T an ideal of R? Is S an ideal of R?

In this question, pA(x) denotes the characteristic polynomial of an n × n matrix A and mA(x) denotes its minimal polynomial. (a)  Working over R, find pA(x) and mA(x) for       0 2 4A= 4 2 0      0 0 3 , which is a 3x3 matrix  (b)  Can the matrix in part (a) be diagonalised over R? Justify your answer. (c)  Repeat parts (a) and (b) with R replaced by F5

16.2.4  Let R=Z[X], and let I be the ideal generated by x3 +2x. Find integers a, b, and c such that in R/I we have I+x5+x6=I+a+bx+cx2.

(8) Let f(x) ∈ Z[x] be an irreducible polynomial of degree 4 such that its Galois group over Q is isomorphic to S4. Let α be a root of f(x). Show that Q(α) has no subfields other than Q and Q(α).

Find the values of k for which A is invertible

Let A ∈ Mn×n(R) be the matrix defined by Aij= 1 for all i and j. Find the characteristic polynomial of A.

If q(x) is an arbitrary polynomial in Pn it follows that                                      q(x) = c0p0(x) + ... + cnpn(x) (1) for some scalars c0, ... , cn                                     (a) Show that ci = q ( ai) for i = 0, ... , n, and deduce that                                  q(x) = q (a0)p0(x) + · · · + q( an)pn(x) is the unique representation of q(x) with respect to the basis ẞ

(a) Prove that if A is invertible and BA = CA, then B=C. (b) Give a counterexample to show that the result in part (a) may fail if A is not invertible.