MATH 113 Study Guide - Midterm Guide: Unique Factorization Domain, Euclidean Domain, Surjective Function
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These problems are practice for the second exam, on rings and elds. Prove that if i is prime, then f is. 3, and consider the eld extensions: q( 2 : q( ) q(i, ) = k. Given that [k : q] = 24, determine [q( ) : q( Consider the extensions q q( ) (b) let = Let f and g be the automorphisms of q( 3 (cid:40) 3 (cid:40) 3 . 1 (c) find an element x q( 3 (d) using (c), and given that [q( 3 (e) prove that gal(q( 3 . 7, ) such that f (g (x)) (cid:54)= g (f (x)). 7, ) : q] = 6, prove that gal(q( 3 . Prove that ker f = (x + 1, x 2 + 1). 5. (a) state the ( rst) isomorphism theorem for rings. (b) consider the map : c[x, y ] c[y ] given by (p(x, y )) = p(y 2, y 3).
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