Homework 3 - Q p(Cauchy sequences mod Cauchy sequences...

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Algebraic number theory (Spring 2013), Homework 3 Frank Thorne, [email protected] Due Monday, February 10 1. (5 points) Represent 23, 1 4 , - 7, and - 1 14 as 7-adic numbers. Which of them are 7-adic integers? 2. (5 points) Write out a formal proof that there exists an injection Z ( p ) Z p . 3. (*7 points) Look up and write out the definition of an inverse limit in general, in terms of a universal property. (For example, see the Wikipedia page.) Prove that Z p is the inverse limit of the rings Z / ( p n ), under the projection morphisms, according to this definition. 4. (5 points) Represent 6 as a 5-adic integer (find the first few 5-adic digits, and prove that you can keep going without quoting Hensel’s lemma), and prove that you cannot represent 6 as a 7-adic integer. 5. (10 points) Starting from the completion definition of
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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 p-adic 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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