Bob Bilson

]]>Hamish Gilmore (who did his Masters with me) found a (5,k)-biclique with Galois groups C(5).

It has 181 edges, 23 vertices, and interesting factor g(x) = x^5-65x^4+1679x^3-21530x^2+136953x-345421.

In fact, he found exactly five quintic number fields F arising from (5,k)-bicliques with Galois group C(5), and whose discriminant D_F has magnitude <= 2 \times 10^7.

See Chapter 7 of Hamish's thesis, available at: http://researchcommons.waikato.ac.nz/handle/10289/9367

Best, Daniel.

]]>For each prime number p, F is a continuous function on the p-adic numbers Q_p. As N is dense in the p-adic integers Z_p, which are closed in Q_p, then F maps Z_p to Z_p, and a fortiori, Z to Z_p.

This holds for any p, and the only rationals which are in Z_p for all p are integers, so F maps Z to Z.

This argument applies to prove that a polynomial mapping a set S to Z maps Z to Z, provided that S is dense in Z_p for all primes p. This is the same as saying that for any integer a and prime power p^k there is some element of S that is congruent to a modulo p^k.

This may seem a “mathematics-made-difficult” style argument, but the idea of using p-adic continuity of polynomials over Q is used in number theory to prove things such as the Clausen-von Staudt congruences for Bernoulli numbers, and that again is the start of the theory of p-adic L-functions.

(Sorry, after my last effort I am giving up using the WordPress latex mode. It is just too frustrating. )

]]>Can’t blame your decision, though.

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There’s some nice combinatorics in this; when is a finite set of elements then both sides count the number of maps from to , the right side splitting up these maps according to the size of their images.

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Nixon fired his Attorney General in 1973.

Nixon was ousted in 1974.

So things are on track.

— Andy Borowitz

Buck up, his term of office may be brief …

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