235HW1 - Math 235: Linear Algebra HW 1 Problem 1.2.1...

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Unformatted text preview: Math 235: Linear Algebra HW 1 Problem 1.2.1 Problem 1.2.1 Verify that the set of complex numbers described in Example 4 is a subfield of C Call the subfield S = x + y √ 2 | x,y ∈ Q We must verify that S meets the following 2 conditions: 1. 0 , 1 ∈ S 2. If a,b ∈ S then a + b,- a,ab and a- 1 ∈ S Throughout let a = x + √ 2 y and b = w + sqrt 2 z where x,y,z,w ∈ Q . First we verify condition 1: • If we set x = y = 0 ∈ Q then we see that a = 0 + √ 2 · 0 = 0 so 0 ∈ S . • If we set x = 1 and y = 0 then we see that a = 1 + √ 2 · 0 so a ∈ S . Now we verify condition 2: • a + b = x + √ 2 y + w + √ 2 z = w + √ 2 z + x + √ 2 y = b + a . • - a =- ( x + √ 2 y ) =- x + √ 2(- y ), since x,y ∈ Q we know that- x,- y ∈ Q thus- a ∈ S . • ab = ( x + √ 2 y )( w + √ 2 z ) = xw + √ 2 xz + √ 2 yw + 2 yz = ( xw + 2 yz ) + √ 2( xz + yw ). And since x,y,z,w ∈ Q we have that ( xw + 2 yz ) ∈ Q and ( xz + yw ) ∈ Q thus ab ∈ S . • Consider r = x x 2- 2 y 2 + √ 2- y x 2- 2 y 2 . Note that x x 2- 2 y 2 ∈ Q and- y x 2- 2 y 2 ∈ Q so r ∈ S . Now consider ( ar = x + √ 2 y ) · x x 2- 2 y 2 + √ 2- y x 2- 2 y 2 = 1 x 2- 2 y 2 ( x + √ 2 y )( x- √ 2 y ) = 1 x 2- 2 y 2 x 2- 2 y 2 = 1 thus a- 1 = r ∈ S . Problem 1.2.8 Prove that each field of characteristic zero contains a copy of the rational number field. First note that any non-zero element of Q can be written as ± a b where a,b ∈ Z + (the positive intergers)....
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This note was uploaded on 11/14/2010 for the course PHYCS 498 taught by Professor Aa during the Spring '10 term at University of Illinois, Urbana Champaign.

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235HW1 - Math 235: Linear Algebra HW 1 Problem 1.2.1...

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