Unformatted text preview: F such that, for every x F , either x R or x-1 R . Prove that the ideals of R are linearly ordered; i.e., if I and J are ideals of R , then either I f J or J f I . 3. Let M 2 ( Q ) be the ring of 2×2 all matrices with rational enties. Prove: a. M 2 ( Q ) has no nontrivial ideals. b. M 2 ( Q ) has an identity but is not a field. FIELDS 1. Find the minimal polynomial for " = over the field of rationals Q and prove it is minimal. 5 2 + 2. Let GF( p n ) denote the Galois field with p n elements. (a) Prove that GF( p a ) f GF( p b ) iff a divides b . (b) Prove that GF( p a ) 1 GF( p b ) = GF( p d ), where d = gcd( a, b ). 3. Let F be a finite field of n = p m elements. Find necessary and sufficient conditions to insure that f ( x ) = x 2 + 1 has a root in F ; i.e., f is not irreducible over F ....
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- Winter '11
- Ring, Characteristic, Cyclic group