MAD 4301 Midterm: MAS4301 F02 Test 3
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Complete the following de nitions. (a) k is a eld extension of f if . (b) i is an ideal of r if . (2) (5 pts. ) State (no proof required) eisenstein"s irreducibility criterion. (3) (35 pts. ) Give examples of the following, or state that none exists. No proofs required. (a) a eld has exactly 2 ideals. (b) in a commutative ring, the set {0} is an ideal. (5) (20 pts. ) Factor the given polynomial into a product of irreducibles in q[x]. (a) f (x) = x7 15x4 + 20x + 30 (b) f (x) = x5 + x2 x 1 (6) (20 pts. ) Let f (x) = x3 + x2 + 2x + 1 q[x]. (a) prove that f (x) is irreducible in q[x]. (b) let c be a root of f (x).
