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Sample Final - for 1 ≤ i ≤ p-1 the binomial coefficeint...

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MATH 113 SAMPLE QUESTIONS 1. Let G be a group. Suppose that A, B are subgroups of G . Let M be the following subset of G , define by: M = { ab | a A, b B } . Show that, if B is a normal subgroup, then the set M is a subgroup of G (in this question, do not assume that G is abelian). 2. Let G be a group, with | G | = p n , for some prime number p , and integer n 1. Let a G be arbitrary. Show that, if G is not cyclic, then a p n - 1 = e. Hint: denote by H = h a i the cyclic subgroup of G generated by a . Use one of the corollaries of Lagrange theorem. 3. Let H = { A GL( n, R ) | det( A ) = 1 } . Show that H is a normal subgroup of GL( n, R ), and we have an isomorphism: GL( n, R ) /H = R * (here the group operation on R * being given by multiplication). Hint: use the fundamental homomorphism theorem. 4. Let p be a prime. Use Eisenstein’s criterion to show that the polynomial f ( x ) = x p - px - 1 is irreducible in Q [ x ] (hint: use the substitution x = y + 1; you can use the result:
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Unformatted text preview: for 1 ≤ i ≤ p-1, the binomial coefficeint ( p i ) is divisible by p ). 5. Fix a prime p . Let S be the following subset of Q , defined by: S = { m n | n not divisible by p } . a) Show that S is a subring of Q , with unity 1. b) Describe the set of units of S . 6. Show that F = { a + bi | a,b ∈ Q } is a subfield of C . 7. Let p be a prime. Let f ( x ) ∈ Z p [ x ]. Let q ( x ) ,r ( x ), be the quotient, and remainder respectively, of f ( x ) upon upon division by the polynomial x p-x . 1 2 MATH 113 SAMPLE QUESTIONS a) Write down the relation between f ( x ) ,q ( x ) ,r ( x ) and x p-x . b) Show that, for α ∈ Z p , we have f ( α ) = 0 mod p if and only if r ( α ) = 0 mod p (hint: Fermat’s theorem)....
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