Unformatted text preview: Q p (Cauchy sequences mod Cauchy sequences converging to zero), prove the following properties, less sketchily than was done in lecture: • Q p is a ﬁeld. • Z p is a ring, and ( p ) is the unique maximal ideal. • Q p and Z p possess an absolute value which agrees with the padic absolute value on Q and Z , and are complete with respect to this absolute value. 6. (5 points) Prove that addition or multiplication by any ﬁxed element of Q p is (topologically) a homeomorphism from Q p to itself. If you want to study valuations in general, the adeles, Tate’s thesis, etc., please be sure to do this exercise. (Or just convince yourself it’s “obvious”.) 7. (7 points) Z p is compact ....
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 Spring '13
 FrankThorne
 Algebra, Number Theory, Integers, Rational number

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