By Kurt Mahler

ISBN-10: 0521231027

ISBN-13: 9780521231022

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**Extra resources for p-adic numbers and their functions**

**Example text**

We shall call the moduli listed above the basic moduli and the characters the basic characters. 39 PRIMITIVE CHARACTERS We can express most of the basic characters, if we limit ourselves to positive values of n, in terms of Jacobi's symbol (mill), which is defined (by multiplying together the corresponding Legendre symbols) when n is odd and positive. We have 3 X4(n) (7) = Xs(n) = X4(n)XS(n) = (~I) = (~4), (~) = (~), (~2) (~8), provided n is odd, which it naturally is when the modulus is 4 or 8.

They are (a) Gauss' theorem on the roots of unity of order q, when q is a prime of the form 2k + 1, and (b) Kummer's problem on the cubic periods. , if q is 3 or 5 or 17 or 257 or 65537), each qth root of unity can be expressed in terms of rational numbers by using a succession of square root signs. )nstruction using ruler and compasses only.

B; and Hence cos 3(} = a 2Jq. This determines cos 3(} except for sign. But the ambiguous sign, arising from the unknown sign of a, can also be specified, for it can be shown that a == 1 (mod 3). To prove this, we consider the number N of solutions of the congruence v3 == u(u + 1) (mod q). For u == 0 or -1 there is just one value of v, and for any other u there are either three values of v or none. Hence N == 2 (mod 3). On the other hand, N = q-l L u=o {l + X[u(u + = q + (A + Bw) 1)] + X2[U(U + 1)]} + (A + Bw 2 ) = q + a.

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