MAD 4301 Midterm: MAS4301 C97 Test 3
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Instructions: do not write your answers on this paper, use separate answer sheets. This is a closed-book test. (1) (4 pts. ) Let f be a eld and f (x), g(x) elements of f [x]. De ne the following. (a) f (x) is irreducible (b) f (x) and g(x) are associates (2) (16 pts. ) Proofs are not required, never- theless, be speci c and describe your examples carefully. (a) a non-abelian group with 6 elements. (b) a eld with 2 elements. A eld with 8 elements. (c) an irreducible polynomial of degree 2 in r[x]. (d) an irreducible polynomial in c[x]. (3) computations. Factor f (x) into a product of irreducibles (i) in q[x] (ii) in z/2[x] (iii) in z/7[x] (b) (10 pts. ) Find 2 irreducible polynomials in z/3[x], both of degree 2, such that they are not associates. (c) (10 pts. ) Find an irreducible polynomial of degree 3 in z/7[x]. (d) (10 pts. )
