Eisenstein's lemma

Builds on Legendre symbol Gauss's lemma on quadratic residues

Statement

Let $p$ be an odd prime and let $a$ be a positive integer with $p \nmid a$ . Then $\left(\frac{a}{p}\right) = (-1)^{\sum_{u} \lfloor a u / p \rfloor},$ where the sum runs over the even integers $u = 2, 4, \dots, p - 1$ , and $\left(\frac{a}{p}\right)$ is the Legendre symbol.

Remarks

For example, take $p = 7$ and $a = 3$ . The sum is $\lfloor 6/7 \rfloor + \lfloor 12/7 \rfloor + \lfloor 18/7 \rfloor = 0 + 1 + 2 = 3$ , so $\left(\frac{3}{7}\right) = -1$ . A common variant holds for odd $a$ : then $\left(\frac{a}{p}\right) = (-1)^{\sum_{k=1}^{(p-1)/2} \lfloor k a / p \rfloor}$ . The proof is similar to the proof of Gauss's lemma. Eisenstein used the lemma in a proof of quadratic reciprocity that counts lattice points in a triangle.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

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Divisibility definition
Congruence modulo n definition
Prime number definition
Quadratic residue definition
Legendre symbol definition
Gauss's lemma on quadratic residues statement
Eisenstein's lemma statement

Tag	`0XSDBCP4RZ`
URI	`m2://0XSDBCP4RZ`
Kind	statement, lemma
Writing	W0 stub
MSC	11A15
Status	This copy of the site has no build data. Run `m2 status 0XSDBCP4RZ`.
Uses	Legendre symbol, Gauss's lemma on quadratic residues

Eisenstein's lemma

Statement

Remarks

Reading path

Further reading

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