MAD 4301 Midterm: MAS4301 S93 Test 4
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Modern algebra final exam april 23, 1993 (1) let f be a eld and p(x) f [x]. Then f [x] (p(x)) is a eld if and only if p(x) is irreducible. Proof: (2) find all rational roots of p(x) = 2x4 3x3 + 3x2 3x + 1. Factor p(x) completely in z[x]. (3) list all irreducible polynomials of degree 3 in z2[x]. (4) give examples. No proofs required. (a) a eld with 2 elements. Then f (x) f (a) mod (x a). , an} and let a be an arbitrary element of g. (a) (a a1, a a2, . , a an) is a permutation of (a1, a2, . Proof: (b) e = a a a (n factors). Proof: (8) de ne the following: (a) f is a eld. (b) g is an abelian group. (c) the polynomial p(x) is irreducible. (9) test for irreduciblity. Let g g. then |g| = p if and only if p 6= e. proof:
