Unformatted text preview: But degrees are nonnegative, so that deg ( f ) = 0 and f ( x ) is a nonzero constant. If R is a f eld, then every nonzero element is a unit in R ; that is, there is v ∈ R with u v = 1. Since R ⊆ R [ x ] , we have v ∈ R [ x ] , and so u is a unit in R [ x ] . (ii) Show that ( [ 2 ] x +[ 1 ] ) 2 = [ 1 ] in I 4 [ x ] . Conclude that the statement in part (i) may be false for commutative rings that are not domains. Solution. ( [ 2 ] x + [ 1 ] ) 2 = [ 4 ] x 2 + [ 4 ] x + [ 1 ] = [ 1 ] in I 4 [ x ] . 3.33 Show that if f ( x ) = x p − x ∈ F p [ x ] , then its polynomial function f [ : F p → F p is identically zero. Solution. Let f ( x ) = x p − x ∈ F p [ x ] . If a ∈ F p , Fermat ’ s theorem gives a p = a , and so f ( a ) = a p − a = 0....
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 Fall '11
 KeithCornell
 Ring, Solution., · · ·, nonzero commutative ring

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