= In this problem you will prove the supplementary case of Quadratic Reciprocity, that is to say to show that () = (1). Let be an odd prime, let s(x) be an irreducible factor of 08(x) in Fp[2], and let E Fp[2]/s(x)Fy[x] where, as in HW8 Q2 we identify Fp C E. Further by HW8 Q2 there exists an element SE E which is a root of s(x) and thus of $8(x). a) Show that 5 # 0 and ordex (5) = 8. b) For T = 5 + 5-1 show that 72 = 2. c) Show that TP (3) d) Show that TP Et with TP = if and only if p? = 1 mod 8. e) Use the above facts to conclude the result. = = 1 = = In this problem you will prove the supplementary case of Quadratic Reciprocity, that is to say to show that () = (1). Let be an odd prime, let s(x) be an irreducible factor of 08(x) in Fp[2], and let E Fp[2]/s(x)Fy[x] where, as in HW8 Q2 we identify Fp C E. Further by HW8 Q2 there exists an element SE E which is a root of s(x) and thus of $8(x). a) Show that 5 # 0 and ordex (5) = 8. b) For T = 5 + 5-1 show that 72 = 2. c) Show that TP (3) d) Show that TP Et with TP = if and only if p? = 1 mod 8. e) Use the above facts to conclude the result. = = 1 =