# 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 ω = e 2 π i / 3 = (- 1 + i √ 3 ) / 2, a primitive cube root of 1, and let R = Z [ ω ] = { a + b ω | 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 + ω + ω 2 = 0, so ω 2 =- 1- ω . 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 ω and y = c + d ω (where a , b , c , d ∈ Z ), then ( a + b ω )( c + d ω ) = ( ac- bd )+( ad + bc- bd ) ω and we see that xy ∈ R . Thus R is a subring of C . Define θ : R → Z 2 × Z 2 by θ ( a + b ω ) = ([ a ] , [ b ]) . Then θ is a group homomorphism (but NOT ring homomorphism) and θ is onto. Furthermore a + b ω ∈ ker θ iff θ ( a + b ω ) = iff ([ a ] , [ b ]) = ([ ] , [ ]) iff a , b ∈ 2 Z iff a + b ω ∈ 2 R . It now follows from the fundamental homomorphism theorem for groups that...
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