Difference between revisions of "Quadratic reciprocity"
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This theorem can help us evaluate Legendre symbols, since the following laws also apply: | This theorem can help us evaluate Legendre symbols, since the following laws also apply: | ||
− | * If <math>a\equiv b\pmod p</math>, then <math>\left(\frac{a}{p}\right)=\left(\frac{b}{p}\right)</math>. | + | * If <math>a\equiv b\pmod{p}</math>, then <math>\left(\frac{a}{p}\right)=\left(\frac{b}{p}\right)</math>. |
* <math>\left(\frac{ab}{p}\right)=\left(\frac{a}{p}\right)\left(\frac{b}{p}\right)</math>. | * <math>\left(\frac{ab}{p}\right)=\left(\frac{a}{p}\right)\left(\frac{b}{p}\right)</math>. | ||
There also exist quadratic reciprocity laws in other [[ring of integers|rings of integers]]. (I'll put that here later if I remember.) | There also exist quadratic reciprocity laws in other [[ring of integers|rings of integers]]. (I'll put that here later if I remember.) |
Revision as of 17:15, 12 July 2006
Let be a prime, and let be any integer not divisible by . Then we can define the Legendre symbol $\left(\frac{a}{p}\right)=
Quadratic Reciprocity Theorem
There are three parts. Let and be distinct odd primes. The the following hold:
- .
- .
- .
This theorem can help us evaluate Legendre symbols, since the following laws also apply:
- If , then .
- .
There also exist quadratic reciprocity laws in other rings of integers. (I'll put that here later if I remember.)