Consider the ring 23 [i] . a] List all the elements of this ring. Be precise with your notation. il'he elements of2'3[1]= {a + bi I11,b E 33} = {0,1,2, 1, 1 +1,2 + i, 21', 1 + 21', 2 + 21'} b} Find all solutions of the quadratic equation 2x2 x + 1 = 0 in the polynomial ring (23 [1])[31]. c] Find all solutions of the quadratic equation 2x2 x + 1 = 0 in the polynomial ring (2'3[1'])[x]. a] List the elements in the ring R = 22 [:I:],I(JC3 + x + 1). Be precise with your notation. There are 23: 8 elements in the ring R = 22 [x]f(x3 + x + 1) The elements are the classes [0], [1], [x], [x+1], [x2], [x2 + 1], [)12 + x], and [x2 + x + 1] b} Find the inverse of [x + 1] in R. If one does not exist, explain why. 1:] Find the inverse of [31:2] in R. If one does not exist, explain why