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Course: MATH 534, Fall 2009
School: Drexel
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Word Count: 250

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2004 Winter DUE: January 20, 2004 Homework One MATH 534 1. Let F be a field with characteristic p &gt; 0. Show that for any elements a, b F we have that (a + b)p = ap + bp . 2. Show that if D is an integral domain with characteristic zero, then D is infinite. 3. Let F be a field with q elements. Show that for all a F we have that aq = a. 4. (a) An element a in a ring R is called nilpotent if for some k 1...

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2004 Winter DUE: January 20, 2004 Homework One MATH 534 1. Let F be a field with characteristic p > 0. Show that for any elements a, b F we have that (a + b)p = ap + bp . 2. Show that if D is an integral domain with characteristic zero, then D is infinite. 3. Let F be a field with q elements. Show that for all a F we have that aq = a. 4. (a) An element a in a ring R is called nilpotent if for some k 1 we have ak = 0. Show that the set of all nilpotent elements in a commutative ring R form a subring of R. (b) Find all nilpotent elements in Z24 . 5. (a) In a ring R an element a R is called idempotent if a2 = a. Show in that an integral domain, 0 and 1 are the only idempotent elements. (b) Find all the idempotent elements in Z6 , Z12 , and Z6 Z12 . 6. Find all possible ring homomorphisms between Z6 and Z10 . 7. For a, b Z, let B(a, b) M (2, Z) be defined by B(a, b) = A = a, b Z}. Show that S Z[ 3] = {a + b 3 : a, b Z}. = a b 3b a . Let S = {B(a, b) : 8. Show that if I is a maximal ideal in Zn if and only if I =< p >, where p is a prime divisor of n. 9. (a) Show that 1 and p are the only elements in the field Zp that are their own inverses. (b) Deduce from part (a) that if p is prime, then (p - 1) ! -1(modp). 1
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