User:Virginia-American/Sandbox/Quadratic Reciprocity

Other rings

There are also quadratic reciprocity laws in rings other than the integers.

Gaussian integers

In his second monograph on biquadratic reciprocity Gauss stated quadratic reciprocity for the ring Z[i] of Gaussian integers, saying that it is a corollary of the biquadratic law in Z[i], but did not provide a proof of either theorem. Dirichlet showed that the law in Z[i] can be deduced from the law for Z without using biquadratic reciprocity. (See the articles on Gaussian integer and biquadratic reciprocity for definitions and notations).

For an odd Gaussian prime π and a Gaussian integer α, gcd(α, π) = 1, define the quadratic character for Z[i] by the formula

${\begin{aligned}\left[{\frac {\alpha }{\pi }}\right]_{2}&=\pm 1\equiv \alpha ^{\frac {\mathrm {N} \pi -1}{2}}{\pmod {\pi }}\\&={\begin{cases}+1{\mbox{ if }}\gcd(\alpha ,\pi )=1{\mbox{ and there is a Gaussian integer }}\beta {\mbox{ such that }}\alpha \equiv \beta ^{2}{\pmod {\pi }}\\-1{\mbox{ if }}\gcd(\alpha ,\pi )=1{\mbox{ and there is no such }}\beta \end{cases}}\end{aligned}}$

Let λ = a + b i and μ = c + d i be distinct Gaussian primes where a and c are odd and b and d are even. Then

${\Bigg [}{\frac {\lambda }{\mu }}{\Bigg ]}_{2}={\Bigg [}{\frac {\mu }{\lambda }}{\Bigg ]}_{2},\;\;\;\;{\Bigg [}{\frac {i}{\lambda }}{\Bigg ]}_{2}=(-1)^{\frac {b}{2}},\;\;{\mbox{ and }}\;\;{\Bigg [}{\frac {1+i}{\lambda }}{\Bigg ]}_{2}={\Bigg (}{\frac {2}{a+b}}{\Bigg )},$

where $({\tfrac {a}{b}})$ is the Jacobi symbol for Z.

Eisenstein integers

The ring of Eisenstein integers is Z[ω], where $\omega ={\frac {-1+{\sqrt {-3}}}{2}}=e^{\frac {2\pi i}{3}}$ is a cube root of 1. (See the articles on Eisenstein integer and cubic reciprocity for definitions and notations).

For an Eisenstein prime π, Nπ ≠ 3 and an Eisenstein integer α, gcd(α, π) = 1, define the quadratic character for Z[ω] by the formula

${\begin{aligned}\left[{\frac {\alpha }{\pi }}\right]_{2}&=\pm 1\equiv \alpha ^{\frac {\mathrm {N} \pi -1}{2}}{\pmod {\pi }}\\&={\begin{cases}+1{\mbox{ if }}\gcd(\alpha ,\pi )=1{\mbox{ and there is an Eisenstein integer }}\beta {\mbox{ such that }}\alpha \equiv \beta ^{2}{\pmod {\pi }}\\-1{\mbox{ if }}\gcd(\alpha ,\pi )=1{\mbox{ and there is no such }}\beta \end{cases}}\end{aligned}}$

Let λ = a + b ω and μ = c + d ω be distinct Eisenstein primes where a and c are not divisible by 3 and b and d are divisible by 3. Eisenstein proved

$\left[{\frac {\lambda }{\mu }}\right]_{2}{\bigg [}{\frac {\mu }{\lambda }}{\bigg ]}_{2}=(-1)^{{\frac {\mathrm {N} \lambda -1}{2}}{\frac {\mathrm {N} \mu -1}{2}}},\;\;\;\;{\bigg [}{\frac {1-\omega }{\lambda }}{\bigg ]}_{2}={\bigg (}{\frac {a}{3}}{\bigg )},\;\;{\mbox{ and }}\;\;{\bigg [}{\frac {2}{\lambda }}{\bigg ]}_{2}={\bigg (}{\frac {2}{\mathrm {N} \lambda }}{\bigg )},$

where $({\tfrac {a}{b}})$ is the Jacobi symbol for Z.

Imaginary quadratic fields

The laws in Z[i] and Z[ω] are special cases of more general laws that hold for the ring of integers in any imaginary quadratic field. Let k be an imaginary quadratic number field with ring of integers ${\mathcal {O}}_{k}=\mathbb {Z} \omega _{1}\oplus \mathbb {Z} \omega _{2},$ where $\left\{\omega _{1},\omega _{2}\right\}$ is an integral basis. For ${\mathfrak {p}}$ a prime ideal of ${\mathcal {O}}_{k}$ with an odd norm and a $\alpha \in {\mathcal {O}}_{k},\;\;\gcd(\alpha ,{\mathfrak {p}})=1,$ define the quadratic character for ${\mathcal {O}}_{k}$ by the formula

${\begin{aligned}\left[{\frac {\alpha }{\mathfrak {p}}}\right]_{2}&=\pm 1\equiv \alpha ^{\frac {\mathrm {N} {\mathfrak {p}}-1}{2}}{\pmod {\mathfrak {p}}}\\&={\begin{cases}+1{\mbox{ if }}\gcd(\alpha ,{\mathfrak {p}})=1{\mbox{ and there is a }}\beta \in {\mathcal {O}}_{k}{\mbox{ such that }}\alpha \equiv \beta ^{2}{\pmod {\mathfrak {p}}}\\-1{\mbox{ if }}\gcd(\alpha ,{\mathfrak {p}})=1{\mbox{ and there is no such }}\beta \end{cases}}\end{aligned}}$

and for $\pi \in {\mathcal {O}}_{k},\;\;(\pi )={\mathfrak {p}}_{1}{\mathfrak {p}}_{2}\dots {\mathfrak {p}}_{n}$ define

${\bigg [}{\frac {\alpha }{\pi }}{\bigg ]}_{2}=\left[{\frac {\alpha }{{\mathfrak {p}}_{1}}}\right]_{2}\left[{\frac {\alpha }{{\mathfrak {p}}_{2}}}\right]_{2}\dots \left[{\frac {\alpha }{{\mathfrak {p}}_{n}}}\right]_{2}$

For $\nu \in {\mathcal {O}}_{k}$ with odd norm Nν, define (ordinary) integers a, b, c, d by the equations

${\begin{aligned}\nu \omega _{1}&=a\omega _{1}+b\omega _{2}\\\nu \omega _{2}&=c\omega _{1}+d\omega _{2}\end{aligned}}$

and define a function χ(ν) where ν has odd norm by

$\chi (\nu )=i^{(b^{2}-a+2)c+(a^{2}-b+2)d+ad}.$

If m = Nμ and n = Nν are both odd, Herglotz proved

${\Bigg [}{\frac {\mu }{\nu }}{\Bigg ]}_{2}\left[{\frac {\nu }{\mu }}\right]_{2}=(-1)^{{\frac {m-1}{2}}{\frac {n-1}{2}}}\chi (\mu )^{m{\frac {n-1}{2}}}\chi (\nu )^{-n{\frac {m-1}{2}}}.$

Also, if $\mu \equiv \mu '{\pmod {4}}{\mbox{ and }}\nu \equiv \nu '{\pmod {4}}$

${\Bigg [}{\frac {\mu }{\nu }}{\Bigg ]}_{2}\left[{\frac {\nu }{\mu }}\right]_{2}={\Bigg [}{\frac {\mu '}{\nu '}}{\Bigg ]}_{2}\left[{\frac {\nu '}{\mu '}}\right]_{2}$

Polynomials over a finite field

Let F be a finite field with q = pⁿ, n ≥ 1 elements, where p is an odd prime number, and let F[x] be the ring of polynomials in one variable with coefficients in F. If $f,g\in \mathrm {F} [x]$ and f is irreducible, monic, and has positive degree, define the quadratic character $({\tfrac {g}{f}})$ for F[x] in the usual manner:

$\left({\frac {g}{f}}\right)={\begin{cases}+1{\mbox{ if }}\gcd(f,g)=1{\mbox{ and there are }}h,k\in \mathrm {F} [x]{\mbox{ such that }}g-h^{2}=kf\\-1{\mbox{ if }}\gcd(f,g)=1{\mbox{ and }}g{\mbox{ is not a square }}{\pmod {f}}\\\;\;\;0{\mbox{ if }}\gcd(f,g)\neq 1\end{cases}}$

If $f=f_{1}f_{2}\dots f_{n}$ is a product of monic irreducibles let

$\left({\frac {g}{f}}\right)=\left({\frac {g}{f_{1}}}\right)\left({\frac {g}{f_{2}}}\right)\dots \left({\frac {g}{f_{n}}}\right).$

Dedekind proved that if $f,g\in \mathrm {F} [x]$ are monic and have positive degrees,

$\left({\frac {g}{f}}\right)\left({\frac {f}{g}}\right)=(-1)^{{\frac {q-1}{2}}(\deg f)(\deg g)}.$

Higher powers

The attempt to generalize quadratic reciprocity for powers higher than the second was one of the main goals that led 19^th century mathemmaticians, inclcuding Gauss, Dirichlet, Jacobi, Eisenstein, Dedekind, and Kummer to the study of general algebraic number fields and their rings or integers, and specifically to Kummer's invention of ideals.

In his famous address to the Paris Congress of Mathematicians in 1900 HIlbert asked for the "Proof of the most general reciprocity law for an arbitrary number field". Artin did so in 1927, building upon work by Takagi, Hasse, and others.

The links below provide more detailed discussions of these theorems.