dec01 - Math 3124 Thursday, December 1 December 1, Ungraded...

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Unformatted text preview: Math 3124 Thursday, December 1 December 1, Ungraded Homework Let = e 2 i / 3 = (- 1 + i 3 ) / 2, a primitive cube root of 1, and let R = Z [ ] = { a + b | a , b Z } . Prove that R is a subring of C and that R / 2 R is a field with 4 elements. Note that 1 + + 2 = 0, so 2 =- 1- . To show that R is a subring of C , we need to show that R is nonempty, and if x , y R , then- x , x + y and xy R . Only the final statement is not obvious. However if x = a + b and y = c + d (where a , b , c , d Z ), then ( a + b )( c + d ) = ( ac- bd )+( ad + bc- bd ) and we see that xy R . Thus R is a subring of C . Define : R Z 2 Z 2 by ( a + b ) = ([ a ] , [ b ]) . Then is a group homomorphism (but NOT ring homomorphism) and is onto. Furthermore a + b ker iff ( a + b ) = iff ([ a ] , [ b ]) = ([ ] , [ ]) iff a , b 2 Z iff a + b 2 R . It now follows from the fundamental homomorphism theorem for groups that...
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