Homework 5 - interesting observations about number fields...

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Algebraic number theory (Spring 2013), Homework 5 Frank Thorne, [email protected] Due Friday, March 22 Recall that starred (*) exercises may involve background beyond what is assumed in this course. 1. (20 points) In a page or so, explain what you learned, and/or some related topics you would like to learn better, from John Voight’s lecture. 2. (5+ points) As we proved, the Minkowski bound gives a lower bound for the discriminant of a number field K in terms of the degree, and also the number of complex embeddings. For each n 6, write out (as a decimal) the Minkowski lower bound for the discriminant of a number field K , and look up (using the Jones-Roberts tables, or otherwise) the smallest discriminant of any number field with that degree. Compare the data. If you have any
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Unformatted text preview: interesting observations about number fields of small discriminant, be sure to share them. 3. (5 points) Compute the class group of Q ( √-33). 4. (5 points) Compute the class group of Q ( √-163). 5. (5 points) Compute the class group of Q ( √-14). 6. (5 points) Compute the class group of Q (7 1 / 3 ). 7. (8 points) Compute the class group of Q ( √ α ), where α 3-α-7 = 0. 8. (8 points) Compute the class group of Q ( ζ 11 ). 9. (*, 12 points) Prove that the class number of Q ( √-13947137572). Hints: No, the Ankeny and Chowla result doesn’t apply. Use Dirichlet’s class number formula, in combination with a computer and some analytic number theory. But just asking PARI or SAGE what the class number is (it’s 17852) is cheating....
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