Deriving modular properties of factors of a polynomial based on the roots of the polynomial.

Question

This is from an answer by Robert to this question When are $n!+1$ and $(n+2)!+1$ coprime?

"The roots of $x^2 + 3x + 1$ are $(-3 \pm \sqrt{5})/2$, so $5$ must be a square mod $p$, which says (if $p > 5$) $p \equiv 1$ or $4 \mod 5$."

Can anyone elaborate on how this is derived or point me somewhere where I can read up on it. Thanks.

There are three assertions in a row in that quote; which one is the "this" you want information on? — Greg Martin, Dec 24 '23 at 02:41

score 1 · Accepted Answer · answered Dec 24 '23 at 02:52

the discriminant of the binary quadratic form $u^2 + 3uv + v^2$ is $9-4=5$

If we have (positive) prime $q$ with Legendre symbol $(5|q) = -1$ and $q | u^2 + 3uv + v^2, \; \; $ then both $q | u$ and $q | v.$ Thus, no such prime $q$ can divide any $x^2 + 3x + 1.$ The number $x^2 + 3x + 1$ may be divisible by $5$ and it may be divisible by primes $p \equiv \pm 1 \pmod 5$

background material includes the Legendre symbol and quadratic reciprocity. Here, we used $(q|5) = (5|q) $ because $5 \equiv 1 \pmod 4$

same thing. Except for $2$ itself, primes are odd, so $1,4 \pmod 5$ refer to $1,9 \pmod{10}$ — Will Jagy, Dec 24 '23 at 03:02

Deriving modular properties of factors of a polynomial based on the roots of the polynomial.

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