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Math8410Ex1.24Jim

# Math8410Ex1.24Jim - Exercise Endler Valuation Theory(3.9...

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Exercise: Endler, Valuation Theory , (3.9). Let ( K, v ) be a non-Archimedean valued field of residue characteristic p > 0 . [Remark: The word ”residue” was omitted when this problem was first posted, which makes the problem almost triv- ial.] Suppose that x and y both lie in the valuation ring. Show that v ( x - y ) > 0 implies v ( x p - y p ) > v ( x - y ) . [stankewicz] Following a hint, let’s recall that x = y + ( x - y ) so x p - y p = ( y + ( x - y )) p - y p = y p + p 1 y p - 1 ( x - y ) + · · · + p p - 1 y ( x - y ) p - 1 + ( x - y ) p - y p = ( x - y ) ( x - y ) p - 1 + p - 1 X i =1 p i y p - i ( x - y ) i - 1 ! = ( x - y ) ( x - y ) p - 1 + p p - 1 X i =1 p - 1 i y p - i ( x - y ) i - 1 ! Define c to be p - 1 i =1 ( p - 1 i ) y p - i ( x - y ) i - 1 and note that c is in the valuation ring (this holds for any commutative ring since we’ve only really used the bi- nomial theorem and the distributive law so far) because x, y and ( p - 1 i ) p - 1 i =1 are. Thus we have shown that x p - y p = ( x - y ) ( ( x - y ) p - 1 + pc ) . Thus by the nonarchimedean assumption
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