Re: Algebraic Number Theory Question (easy?)
From: Phil Carmody (thefatphil_demunged_at_yahoo.co.uk)
Date: 01/21/05
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Date: 21 Jan 2005 04:02:31 +0200
woolcutt@gmail.com writes:
> I'm probably just being lazy, but could anyone give me a hand with
> understanding this brief outline of why prime numbers p congruent to 1
> mod 4 are the sums of two squares:
>
> p is a prime number congruent to 1 mod 4, so -1 is a quadratic residue
> mod p.
> Therefore there is an x so that p divides x^2 + 1.
> Factor x^2 + 1 as x^2 + 1 = (x + i)(x - i) (so good so far; next line
> is fuzzy for me)
> "Since p can devide neither factor, it follows that p is not prime in
> Z[i]"
>
> The author hasn't demonstrated why p doesn't divide either factor
> explicityly, and this is basically an outline of his explanation, but I
> assume there is some easy point that I'm overlooking.
What's your definition of "prime". If you see no distinction between
"prime" and "irreducible", then get a better definition of a prime.
Phil
-- The answer to life's mystery is simple and direct: Sex and death. -- Ian 'Lemmy' Kilminster.
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