Kronecker symbol

<http://www.wikidata.org/entity/Q1789847>

generalization of the Jacobi symbol to all integers

rdfs:label	"Kronecker symbol"@en "Kronecker-Symbol"@de "Kronecker-symbool"@nl "Simbolo di Kronecker"@it "Symbole de Kronecker (théorie des nombres)"@fr "Símbol de Kronecker"@ca "simbolo de Kronecker"@eo "símbolo de Kronecker"@es "símbolo de Kronecker"@pt "символ Кронекера — Якоби"@ru "символ Кронекера — Якобі"@uk "סימן קרונקר"@he "ခရိုနက်ကာ သင်္ကေတ"@my "クロネッカーの記号"@ja "克罗内克符号"@zh "크로네커 기호"@ko
skos:altLabel	"Kroneckersymbol"@de "simbolo de Kronecker"@es "символ Кронекера"@ru "символ Кронекера"@uk "символ Кронекера-Якоби"@ru "クロネッカー記号"@ja
schema:description	"Símbolo matemático"@pt "Verallgemeinerung des Jacobi-Symbols in der Zahlentheorie"@de "generalization of the Jacobi symbol to all integers"@en "야코비 기호를 임의의 정수로 확장한 함수"@ko
schema:dateModified	"2026-09-06T13:44:41+00:00"^^xsd:dateTime
schema:version	2541102577
wikibase:identifiers	4
wikibase:sitelinks	12
wikibase:statements	12
p:P12888	wd:statement/Q1789847-609C5382-0527-46F2-B9AA-CA3E0129B250
p:P138	wd:statement/Q1789847-97bbf5f7-4a81-b220-ea32-20b2de0cb6be
p:P170	wd:statement/Q1789847-a877838c-4d25-571f-969d-2479f569388d
p:P2534	wd:statement/Q1789847-05e0c5af-4c0d-5aa6-d1fa-ba5df2bfaac4
p:P2579	wd:statement/Q1789847-fa365f42-4536-1879-a5a4-c4c033feee0b
p:P2812	wd:statement/Q1789847-2D6A297F-2EAA-44E7-BE74-2F2D1172D2C3
p:P31	wd:statement/Q1789847-48b5733a-479e-a921-0dba-2338252b59f3
	wd:statement/Q1789847-9b54977d-4338-4302-656a-9f38e98e6a42
p:P571	wd:statement/Q1789847-549af3c1-44aa-0e8e-490d-608621ec57ba
p:P6104	wd:statement/Q1789847-460A6483-3FD3-46ED-BF82-D301462A8152
p:P6366	wd:statement/Q1789847-C6B138E7-456C-492F-80B6-DE5A87CBCD46
p:P646	wd:statement/Q1789847-D22D36CC-4429-48E5-8BDB-41ADE030DC5C
p:direct-normalized/P6366	<https://makg.org/entity/177217850>
wdt:P12888	"df-lgs"
wdt:P138	Leopold Kronecker
wdt:P170	Leopold Kronecker
wdt:P2534	"<math class="mwe-math-element mwe-math-element-inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow data-mjx-texclass="ORD"><mtable columnalign="right left right left right left right left right left right left" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true" rowspacing="3pt"><mtr><mtd><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>a</mi></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>u</mi><munder><mo stretchy="false">∏</mo><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></munder><msubsup><mi>p</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow><mrow data-mjx-texclass="ORD"><msub><mi>e</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub></mrow></msubsup></mrow></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd><mtd><mo stretchy="false">=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>a</mi></mrow><mrow data-mjx-texclass="ORD"><mi>u</mi></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><munder><mo stretchy="false">∏</mo><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></munder><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>a</mi></mrow><mrow data-mjx-texclass="ORD"><msub><mi>p</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><msub><mi>e</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub></mrow></msup></mtd></mtr><mtr><mtd><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>a</mi></mrow><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd><mtd><mo stretchy="false">=</mo><mo stretchy="false">[</mo><mi>a</mi><mo stretchy="false">=</mo><mo stretchy="false">±</mo><mn>1</mn><mo stretchy="false">]</mo></mtd></mtr><mtr><mtd><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>a</mi></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo stretchy="false">−</mo><mn>1</mn></mrow></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd><mtd><mo stretchy="false">=</mo><mrow data-mjx-texclass="ORD"><mo data-mjx-texclass="OPEN">{</mo><mtable columnspacing="1em" rowspacing="4pt" columnalign="left left"><mtr><mtd><mn>1</mn></mtd><mtd><mi>a</mi><mo stretchy="false">≥</mo><mn>0</mn></mtd></mtr><mtr><mtd><mo stretchy="false">−</mo><mn>1</mn></mtd><mtd><mi>a</mi><mo><</mo><mn>0</mn></mtd></mtr></mtable><mo fence="true" stretchy="true" symmetric="true" data-mjx-texclass="CLOSE"></mo></mrow></mtd></mtr><mtr><mtd><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>a</mi></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd><mtd><mo stretchy="false">=</mo><mrow data-mjx-texclass="ORD"><mo data-mjx-texclass="OPEN">{</mo><mtable columnspacing="1em" rowspacing="4pt" columnalign="left left"><mtr><mtd><mn>0</mn></mtd><mtd><mn>2</mn><mo stretchy="false">\|</mo><mi>a</mi></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mi>a</mi><mo stretchy="false">≡</mo><mo stretchy="false">±</mo><mn>1</mn><mrow data-mjx-texclass="ORD"><mspace width="0.444em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mn>8</mn><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr><mtd><mo stretchy="false">−</mo><mn>1</mn></mtd><mtd><mi>a</mi><mo stretchy="false">≡</mo><mo stretchy="false">±</mo><mn>3</mn><mrow data-mjx-texclass="ORD"><mspace width="0.444em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mn>8</mn><mo stretchy="false">)</mo></mrow></mtd></mtr></mtable><mo fence="true" stretchy="true" symmetric="true" data-mjx-texclass="CLOSE"></mo></mrow></mtd></mtr><mtr><mtd><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>a</mi></mrow><mrow data-mjx-texclass="ORD"><mi>p</mi></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd><mtd><mo stretchy="false">≡</mo><msup><mi>a</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">−</mo><mn>1</mn><mo stretchy="false">)</mo><mo lspace="0" rspace="0">/</mo><mn>2</mn></mrow></mrow></msup><mrow data-mjx-texclass="ORD"><mspace width="0.444em"></mspace><mo stretchy="false">(</mo><mi>mod</mi><mspace width="0.333em"></mspace><mi>p</mi><mo stretchy="false">)</mo></mrow></mtd></mtr></mtable></mrow></mstyle></mrow></math>"^^http://www.w3.org/1998/Math/MathML
wdt:P2579	number theory
wdt:P2812	"KroneckerSymbol"
wdt:P31	function
	mathematical concept
wdt:P571	"1885-01-01T00:00:00+00:00"^^xsd:dateTime
wdt:P6104	WikiProject Mathematics
wdt:P6366	"177217850"
wdt:P646	"/m/034cx4"

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