Adv Alegbra HW Solutions 108

Adv Alegbra HW Solutions 108 - 108 If x , y R and x = 0,...

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108 If x , y R and x ±= 0, there are q , r R with y = qx + r , where either r = 0o r ∂( r )<∂ ( x ) . In the second case, it follows that 0 ( r )<∂ 0 ( x ) ,for m 1 > 0. Hence, 0 is also a degree function on R . 3.69 Let R be a Euclidean ring with degree function . (i) Prove that ∂( 1 ) ∂( a ) for all nonzero a R . Solution. By the f rst axiom in the de f nition of degree, ∂( 1 ) ∂( 1 · r ) = ∂( r ). (ii) Prove that a nonzero u R is a unit if and only if ∂( u ) = ∂( 1 ) . Solution. If u is a unit, then there is v R with u v = 1; therefore, ∂( u ) ∂( u v) = ∂( 1 ) . By part (i), ∂( u ) = ∂( 1 ) . Conversely, assume that ∂( u ) = ∂( 1 ) . By the division algorithm in R , there are q , r R with 1 = qu + r , where either r = 0or ∂( r )<∂ ( u ) .B u t ∂( u ) = ∂( 1 ) and, by part (i), the possibility ∂( r )<∂( u ) cannot occur. Therefore, r = 0 and u is a unit, for 1 = qu . 3.70 Let α = 1 2 ( 1 + 19 ) , and let R = Z [ α ] . (i) Prove that N : R × N ,de f ned by N ( m + n α) = m 2 mn + 5 n 2 , is multiplicative: N ( u v) = N ( u
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This note was uploaded on 12/21/2011 for the course MAS 4301 taught by Professor Keithcornell during the Fall '11 term at UNF.

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