provide a full handwritten solution to all parts

2. RRT and Tower of fields (from chapters 8 and 12) a) Prove W a Qh/E). b) Looking for a fieid extension of Q that contains #5 naturally we consider (QR/5] = {a + {AVE : ri. b E Q}. Prove that (QR/E] is not a field. c) Read Definition 12216. If possible. find a shortest tower of fields such that the final field contains W. and if not. explain why it is impossible. d) Determine a tower of fields in which the polynomial :52 x/3:r; 1 has a root. e) Prove using RRT that the polynomial :53 + x + 1 has no rational roots. (Hint: RRT does not apply if the coefficients of the polynomial are not rational. You need to transform the polynomial somehow.)