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Unformatted text preview: for 1 ≤ i ≤ p1, 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 px . 1 2 MATH 113 SAMPLE QUESTIONS a) Write down the relation between f ( x ) ,q ( x ) ,r ( x ) and x px . 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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 Fall '08
 OGUS
 Math, Algebra, Group Theory, Normal subgroup, Prime number, Cyclic group

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