# Henry B. Mann.'s Addition theorems; the addition theorems of group theory and PDF

February 27, 2018 | | By admin | By Henry B. Mann.

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9. Let the polynomial F(x" ... , x n ) have coefficients in 0. and, denote by C m (m ;:> 0) the number ()f solutions to the congruence F(x" ... , x n ) ~ 0 (mod pm). '=o Cmf m. It has been conjectured that the series cpU), called the Poincare series of the polynomial F, represems a rational function of f. Find the Poincare series for the polynomial F = E,X,2 + ... + EnX/, where E, is a p-aaic unit, and check that the function cp(f) is rational. 10. Find the Poincare series for a pOlynomial F(x" ...

Show that if m and p are relatively prime, then any p-adic unit E, satisfying the congruence E =; 1 (mod p), is an mth power in R p • 2. Let m = lma, (ma, p) = I, and let E ==: 1 (mod p2 6+1). Show that the p-adic unit E is an mth power in R p • Sec. 5] QUADRATIC FORMS WITH p-ADIC COEFFICIENTS 47 3. If p # 2 and the p-adic integers

Es {x. } and. } from the ciasses a and fl. 36 CONGRUENCES [Chap. 1 It is easily verified that I<: is a ring with unit. Zero and one are the classes containing the sequences {O, 0, ... } and {l, 1, ... }. We now show that I<: is a field. If a is a nonzero class, and {x n } is a Cauchy sequence in this class, then, from some point on (say for n ~ no), all X n are different from zero. Consider the sequence {Yn}, defined by for n < no, for It is easily shown that the sequence {Yn} is a Cauchy sequence, and that its class is the inverse of a.