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home5 - c = b x = x 1 x 2 x k ∈ R and lim k →∞ x k...

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MATH 4107 HOMEWORK #5 DUE: April 12, 2010 Work the following problems and hand in your solutions. You may work together with other people in the class, but you must each write up your solutions independently. A subset of these will be selected for grading. Write LEGIBLY on the FRONT side of the page only, and STAPLE your pages together. 1. 3.2 #13. Hint: If α 1 , . . . , α k are disjoint cycles, what is the order of the composition α 1 · · · α k ? (We discussed this in class.) 2. 3.3 #6. Hint: If you look at the composition of two transpositions, there are three possibilities: (a) ( i j )( k ℓ ) with i , j , k , ℓ distinct, (b) ( i j )( i ℓ ) with i , j , ℓ distinct, (c) ( i j )( i j ) with i , j distinct. 3. Prove that R = { a + b √ 2 : a, b ∈ Q } is a field under the operations of addition and multiplication of real numbers. 4. Define c and c 0 to be the following sets of infinite sequences of real numbers:
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Unformatted text preview: c = b x = ( x 1 , x 2 , . . . ) : x k ∈ R and lim k →∞ x k exists B , c = b x = ( x 1 , x 2 , . . . ) : x k ∈ R and lim k →∞ x k = 0 B . You may assume without proof that these are both commutative rings under the following operations: if x = ( x 1 , x 2 , . . . ) and y = ( y 1 , y 2 , . . . ) then x + y = ( x 1 + y 1 , x 2 + y 2 , . . . ) and xy = ( x 1 y 1 , x 2 y 2 , . . . ) . (a) Determine (with proof) whether c or c has a multiplicative identity. (b) Show directly that c is a (two-sided) ideal in c . (c) Use the First Homomorphism Theorem to show that c/c ∼ = R . (d) Given x ∈ c , give an explicit description of the coset x + c . 1...
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