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Q: Elementary Number Theory ( No Answer,   1 Comment )
Question  
Subject: Elementary Number Theory
Category: Science > Math
Asked by: adorvis-ga
List Price: $100.00
Posted: 01 Dec 2005 11:48 PST
Expires: 19 Dec 2005 16:49 PST
Question ID: 600152
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.
Answer  
There is no answer at this time.

Comments  
Subject: Re: Elementary Number Theory
From: etotheipi-ga on 14 Dec 2005 17:30 PST
 
For the second problem:
Suppose that p is not congruent to 1 (mod q). From Fermat's theorem,
we get p^(q-1)=1(mod q). p is also a primitive root in mod q.
Therefore, q-1|p. However p is a prime number. Therefore, q-1=p and
q=1 (mod p).

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