Reciprocity law

${\displaystyle f(x)\equiv f_p(x)=(x-n_p)(x-m_p)(\text{mod }p)}$

holds. For a general reciprocity law^{pg 3}, it is defined as the rule determining which primes ${\displaystyle p}$ the polynomial ${\displaystyle f_p}$ splits into linear factors, denoted ${\displaystyle \text{Spl}\{f(x)\}}$.

There are several different ways to express reciprocity laws. The early reciprocity laws found in the 19th century were usually expressed in terms of a power residue symbol (p/q) generalizing the quadratic reciprocity symbol, that describes when a prime number is an nth power residue modulo another prime, and gave a relation between (p/q) and (q/p). Hilbert reformulated the reciprocity laws as saying that a product over p of Hilbert norm residue symbols (a,b/p), taking values in roots of unity, is equal to 1. Artin reformulated the reciprocity laws as a statement that the Artin symbol from ideals (or ideles) to elements of a Galois group is trivial on a certain subgroup. Several more recent generalizations express reciprocity laws using cohomology of groups or representations of adelic groups or algebraic K-groups, and their relationship with the original quadratic reciprocity law can be hard to see.

The name reciprocity law was coined by Legendre in his 1785 publication Recherches d'analyse indéterminée, because odd primes reciprocate or not in the sense of quadratic reciprocity stated below according to their residue classes ${\displaystyle mod4}$. This reciprocating behavior does not generalize well, the equivalent splitting behavior does. The name reciprocity law is still used in the more general context of splittings.

Quadratic reciprocity

In terms of the Legendre symbol, the law of quadratic reciprocity states

for positive odd primes <span class="mwe-math-element" typeof="mw:Extension/math" about="#mwt39" data-mw='{"name":"math","attrs":{},"body":{"extsrc":"p,q"}}' id="mwPw"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p,q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p,q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/953a97b9fe7d257c9666fb3cf6bf75380295e2cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:3.362ex; height:2.009ex;" alt="{\displaystyle p,q}"/></span> we have <span class="mwe-math-element" typeof="mw:Extension/math" about="#mwt40" data-mw='{"name":"math","attrs":{},"body":{"extsrc":" \\left(\\frac{p}{q}\\right) \\left(\\frac{q}{p}\\right) = (-1)^{\\frac{p-1}{2}\\frac{q-1}{2}}."}}' id="mwQA"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {p}{q}}\right)\left({\frac {q}{p}}\right)=(-1)^{{\frac {p-1}{2}}{\frac {q-1}{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>q</mi> <mi>p</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>p</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>q</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {p}{q}}\right)\left({\frac {q}{p}}\right)=(-1)^{{\frac {p-1}{2}}{\frac {q-1}{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/056c41d860587e6b676056d15ea25d3d054e3120" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.366ex; height:6.176ex;" alt="{\displaystyle \left({\frac {p}{q}}\right)\left({\frac {q}{p}}\right)=(-1)^{{\frac {p-1}{2}}{\frac {q-1}{2}}}.}"/></span>

Using the definition of the Legendre symbol this is equivalent to a more elementary statement about equations.

For positive odd primes <span class="mwe-math-element" typeof="mw:Extension/math" about="#mwt49" data-mw='{"name":"math","attrs":{},"body":{"extsrc":"p,q"}}' id="mwQw"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p,q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p,q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/953a97b9fe7d257c9666fb3cf6bf75380295e2cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:3.362ex; height:2.009ex;" alt="{\displaystyle p,q}"/></span> the solubility of <span class="mwe-math-element" typeof="mw:Extension/math" about="#mwt50" data-mw='{"name":"math","attrs":{},"body":{"extsrc":"n^2-p\\equiv0\\bmod q"}}' id="mwRA"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{2}-p\equiv 0{\bmod {q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−<!-- − --></mo> <mi>p</mi> <mo>≡<!-- ≡ --></mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo lspace="thickmathspace" rspace="thickmathspace">mod</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{2}-p\equiv 0{\bmod {q}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89dfaf9a058ffcbecbb74bff6a4d34665a0e2cb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.47ex; height:3.009ex;" alt="{\displaystyle n^{2}-p\equiv 0{\bmod {q}}}"/></span> for <span class="mwe-math-element" typeof="mw:Extension/math" about="#mwt51" data-mw='{"name":"math","attrs":{},"body":{"extsrc":"n"}}' id="mwRQ"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"/></span> determines the solubility of <span class="mwe-math-element" typeof="mw:Extension/math" about="#mwt52" data-mw='{"name":"math","attrs":{},"body":{"extsrc":"m^2-q\\equiv0\\bmod p"}}' id="mwRg"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m^{2}-q\equiv 0{\bmod {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−<!-- − --></mo> <mi>q</mi> <mo>≡<!-- ≡ --></mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo lspace="thickmathspace" rspace="thickmathspace">mod</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m^{2}-q\equiv 0{\bmod {p}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79e22654e78c445cffe7102f6dfaaf76fc010112" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.116ex; height:3.009ex;" alt="{\displaystyle m^{2}-q\equiv 0{\bmod {p}}}"/></span> for <span class="mwe-math-element" typeof="mw:Extension/math" about="#mwt53" data-mw='{"name":"math","attrs":{},"body":{"extsrc":"m"}}' id="mwRw"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"/></span> and vice versa by the comparatively simple criterion whether <span class="mwe-math-element" typeof="mw:Extension/math" about="#mwt54" data-mw='{"name":"math","attrs":{},"body":{"extsrc":"(-1)^{\\frac{p-1}{2}\\frac{q-1}{2}}"}}' id="mwSA"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-1)^{{\frac {p-1}{2}}{\frac {q-1}{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>p</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>q</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-1)^{{\frac {p-1}{2}}{\frac {q-1}{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bd23b31d27dc3c19154fcb5c6848b7d82c35e61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.38ex; height:4.343ex;" alt="{\displaystyle (-1)^{{\frac {p-1}{2}}{\frac {q-1}{2}}}}"/></span> is <span class="mwe-math-element" typeof="mw:Extension/math" about="#mwt55" data-mw='{"name":"math","attrs":{},"body":{"extsrc":"1"}}' id="mwSQ"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"/></span> or <span class="mwe-math-element" typeof="mw:Extension/math" about="#mwt56" data-mw='{"name":"math","attrs":{},"body":{"extsrc":"-1"}}' id="mwSg"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−<!-- − --></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/704fb0427140d054dd267925495e78164fee9aac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}"/></span>.

By the factor theorem and the behavior of degrees in factorizations the solubility of such quadratic congruence equations is equivalent to the splitting of associated quadratic polynomials over a residue ring into linear factors. In this terminology the law of quadratic reciprocity is stated as follows.

For positive odd primes <span class="mwe-math-element" typeof="mw:Extension/math" about="#mwt63" data-mw='{"name":"math","attrs":{},"body":{"extsrc":"p,q"}}' id="mwTw"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p,q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p,q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/953a97b9fe7d257c9666fb3cf6bf75380295e2cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:3.362ex; height:2.009ex;" alt="{\displaystyle p,q}"/></span> the splitting of the polynomial <span class="mwe-math-element" typeof="mw:Extension/math" about="#mwt64" data-mw='{"name":"math","attrs":{},"body":{"extsrc":"x^2-p"}}' id="mwUA"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}-p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−<!-- − --></mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}-p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2cd704ec8f5f352ad29584ae5d9e5658e67c311" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.394ex; height:3.009ex;" alt="{\displaystyle x^{2}-p}"/></span> in <span class="mwe-math-element" typeof="mw:Extension/math" about="#mwt65" data-mw='{"name":"math","attrs":{},"body":{"extsrc":"\\bmod q"}}' id="mwUQ"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bmod {q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo lspace="thickmathspace" rspace="thickmathspace">mod</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bmod {q}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54984c2a264427744970463a2a62b145a7a9d212" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.751ex; height:2.509ex;" alt="{\displaystyle {\bmod {q}}}"/></span>-residues determines the splitting of the polynomial <span class="mwe-math-element" typeof="mw:Extension/math" about="#mwt66" data-mw='{"name":"math","attrs":{},"body":{"extsrc":"x^2-q"}}' id="mwUg"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}-q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−<!-- − --></mo> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}-q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71c9fae72ecdfc84a0f9ed07bf31038417263be4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.294ex; height:3.009ex;" alt="{\displaystyle x^{2}-q}"/></span> in <span class="mwe-math-element" typeof="mw:Extension/math" about="#mwt67" data-mw='{"name":"math","attrs":{},"body":{"extsrc":"\\bmod p"}}' id="mwUw"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bmod {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo lspace="thickmathspace" rspace="thickmathspace">mod</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bmod {p}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1460330b9d39cce63f1625732fd347e4a92fda9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.851ex; height:2.509ex;" alt="{\displaystyle {\bmod {p}}}"/></span>-residues and vice versa through the quantity <span class="mwe-math-element" typeof="mw:Extension/math" about="#mwt68" data-mw='{"name":"math","attrs":{},"body":{"extsrc":"(-1)^{\\frac{p-1}{2}\\frac{q-1}{2}}\\in\\{\\pm 1\\}"}}' id="mwVA"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-1)^{{\frac {p-1}{2}}{\frac {q-1}{2}}}\in \{\pm 1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>p</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>q</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> <mo>∈<!-- ∈ --></mo> <mo fence="false" stretchy="false">{</mo> <mo>±<!-- ± --></mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-1)^{{\frac {p-1}{2}}{\frac {q-1}{2}}}\in \{\pm 1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7aeef96496c766d48500b21355ba02228d1578f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.517ex; height:4.343ex;" alt="{\displaystyle (-1)^{{\frac {p-1}{2}}{\frac {q-1}{2}}}\in \{\pm 1\}}"/></span>.

This establishes the bridge from the name giving reciprocating behavior of primes introduced by Legendre to the splitting behavior of polynomials used in the generalizations.