There are actually three questions here. If you are only able to
answer one or two of them that is fine I will just edit the question
appropriatly.
These are standard results from Number Theory that I would like
explained (in detail) or proved.
First,
Let p be a prime such that p is congruent to 3 (mod 4). Then the
polynomial x^4+1 is the product of two irreducible factors in F_p[X]
(polynomials with coefficients in the field with p elements).
Second,
If p and q are primes with p^p congruent to 1 (mod q) then p is
congruent to 1 (mod q) or q is congruent to 1 (mod p). Furthermore,
at least one of the prime factors of (p^p)-1 is congruent to 1 (mod
p).
Third,
Let p be an odd prime. Let a_1, a_2, ..., a_n, b be elements in F_p
(the field with p elements) with n >= 2 (greater than or equal to) and
a_1, ..., a_n non-zero. Give an explicit formula for the number N of
solutions to the equation a_1(x_1)^2 + a_2(x_2)^2 + ... + a_n(x_n)^2 =
b in F_p. Also show that N is not zero and that 2*ord_p(N)>=n-2.
Thanks, if you need any clarification just let me know. |