Exercise:Endler,Valuation Theory, (3.9).Let(K, v)be a non-Archimedeanvalued field of residue characteristicp >0. [Remark: The word ”residue” wasomitted when this problem was first posted, which makes the problem almost triv-ial.] Suppose thatxandyboth lie in the valuation ring. Show thatv(x-y)>0impliesv(xp-yp)> v(x-y).[stankewicz] Following a hint, let’s recall thatx=y+ (x-y) soxp-yp=(y+ (x-y))p-yp=yp+p1yp-1(x-y) +· · ·+pp-1y(x-y)p-1+ (x-y)p-yp=(x-y)(x-y)p-1+p-1Xi=1piyp-i(x-y)i-1!=(x-y)(x-y)p-1+pp-1Xi=1p-1iyp-i(x-y)i-1!Definecto be∑p-1i=1(p-1i)yp-i(x-y)i-1and note thatcis in the valuationring (this holds for any commutative ring since we’ve only really used the bi-nomial theorem and the distributive law so far) becausex, yand(p-1i)p-1i=1are.Thus we have shown thatxp-yp= (x-y)((x-y)p-1+pc). Thus by thenonarchimedean assumption
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