oct24 - ( x ) = 0 (as a polynomial in y with coefcients in...

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Math 5125 Monday, October 24 October 24, Ungraded Homework Let k be an integral domain and let f , g k [ x , y , z ] . Suppose fg k [ x ] . Prove that f , g k [ x ] . Suppose the result is not true. Then without loss of generality, we may assume that a positive power of y appears in f , in other words f k [ x , z ][ y ] and deg ( f ) > 0. Since k [ x , z ] is an integral domain (polynomial rings over integral domains are integral domains) and deg
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Unformatted text preview: ( x ) = 0 (as a polynomial in y with coefcients in k [ x , z ] ), the formula deg ( fg ) = deg ( f ) + deg ( g ) shows that deg ( g ) is a negative integer, which is impossible. The result follows....
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