question is complete.

Consider E= (13 _5) . Let a = [x] E E. (a) Show that E is a field. (b) Find a, b, c E Q such that (a + bo + co?) (1 + @) =1 e E, i.e. a multiplicative inverse for 1 + a e E. (This is possible because E is a field, and according to a theorem in class, every element of E can be represented as a Q-linear combination of 1, a, 02.) (c) Check your answer in part (b) numerically (e.g. using a calculator) by comparing 1 1+ 35 and a + bv5 + c(15)' ER. (E is isomorphic to the subfield Q(v5) = {a + by5 + c(v5)2 : a, b,cc Q} CR