Lagmental Vicfred

Gauss's Lemma Counts Sign Crossings by Vicfred

tags: mathematics, quadratic residues, quadratic reciprocity, sums of squares

The Legendre symbol is minus one to the number of least residues of a, 2a, and so on that exceed p over two. I will separate the object being defined from the consequence being claimed.

Start locally

For an odd prime \(p\), the Legendre symbol \(\left(\frac ap\right)\) records whether \(a\) is a nonzero square modulo \(p\). Reciprocity exchanges numerator and denominator up to a sign.

$$ N=\#\left\{1\le k\le\frac{p-1}{2}:\langle ka\rangle_p>\frac p2\right\} $$

I keep the defining relation \(\mathsf D\) above the derived relation \(\mathsf C\). This exposes whether cancellation used \(x\ne0\) and whether the conclusion is canonical.

$$ \left(\frac ap\right)=(-1)^N $$

Compute before generalising

An explicit case prevents the notation from becoming ceremonial. Every subscript and superscript in the display contributes to the value.

$$ p=7,\ a=3:\quad(3,6,2),\quad N=1,\quad\left(\frac37\right)=-1 $$

The global view

The invariant statement is the one that does not depend on a convenient choice of coordinates, representatives, basis, or enumeration.

$$ \begin{aligned} \mathsf{D}\;&:\quad N=\#\left\{1\le k\le\frac{p-1}{2}:\langle ka\rangle_p>\frac p2\right\},\\[5pt] \mathsf{C}\;&:\quad \left(\frac ap\right)=(-1)^N. \end{aligned} $$

Edge conditions

The symbol is defined modulo an odd prime and is not ordinary division. Composite odd denominators require the Jacobi symbol, which can equal one without certifying a square.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \left(\frac ap\right)=(-1)^N \end{gathered}} $$

The important habit is to remember what was fixed before the calculation began and what was proved only afterward. The final display preserves that order.

This article was posted on Wed 20 November 2024. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.