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==Abstract algebra==
===ModularAritmetika arithmeticmodular anddan finiteMedan fieldsberhingga===
{{Main|ModularAritmetika arithmeticmodular}}
ModularAritmetika arithmeticmodular modifiesmemodifikasi usualaritmetika arithmeticbiasa, byhanya onlysaja usingdengan themenggunakan numbersbilangan <math>\{0,1,2,\dots,n-1\}</math>, for auntuk naturalbilangan numberasli <math>n</math> calledyang thedisebut modulus.
AnyBilangan otherasli naturallainnya numberdapat candipetakan beke mappeddalam intosistem thisini systemdengan bymenggantinya replacingdengan itsisa bysetelah itspembagian remainder after division bydengan <math>n</math>.<ref>{{harvtxt|Kraft|Washington|2014}}, [https://books.google.com/books?id=VG9YBQAAQBAJ&pg=PA96 PropositionProposisi 5.3], phal. 96.</ref>
onJumlah, thepembeda, resultdan ofdarab themodular usualdihitung sumdengan melakukan penggantian yang sama dengan sisa hasil penjumlahan, differenceselisih, oratau perkalian productbilangan ofbulat integersbiasa.<ref>{{cite book|title=Algebra in Action: A Course in Groups, Rings, and Fields|volume=27|series=Pure and Applied Undergraduate Texts|first=Shahriar|last=Shahriari|author-link= Shahriar Shahriari |publisher=American Mathematical Society|year=2017|isbn=978-1-4704-2849-5|pages=20–21|url=https://books.google.com/books?id=GJwxDwAAQBAJ&pg=PA20}}</ref> EqualityKesamaan ofbilangan integersbulat correspondssesuai todengan '' congruencekongruensi'' indalam aritmetika modular arithmetic: ▼Modular sums, differences and products are calculated by performing the same replacement by the remainder
<math>x</math> dan <math>y</math> adalah kongruen (ditulis <math>x\equiv y</math> mod <math>n</math>) ketika mereka memiliki sisa yang sama setelah dibagi dengan <math>n</math>.<ref>{{harvnb|Dudley|1978}}, [https://books.google.com/books?id=tr7SzBTsk1UC&pg=PA28 Teorema 3, hal. 28].</ref> Namun, dalam sistem bilangan ini, [[Pembagian (matematika)|pembagian]] dengan semua bilangan bukan nol dimungkinkan jika dan hanya jika modulusnya adalah prima. Misalnya, dengan bilangan prima <math>7</math> sebagai modulus, pembagian dengan <math>3</math> adalah dimungkinkan: <math>2/3\equiv 3\bmod{7}</math> karena kemungkinan [[menghapus penyebut]] dengan mengalikan kedua ruas dengan <math>3</math> diberikan rumus yang valid <math>2\equiv 9\bmod{7}</math>. Namun, dengan modulus komposit <math>6</math>, pembagian dengan <math>3</math> adalah hal mustahil. Tidak ada solusi yang valid untuk <math>2/3\equiv x\bmod{6}</math>: menghapus penyebut dengan mengalikan dengan <math>3</math> menyebabkan ruas kiri menjadi <math>2</math> sedangkan ruas kanan menjadi <math>0</math> atau <math>3 </math>. Dalam terminologi [[aljabar abstrak]], kemampuan untuk melakukan pembagian berarti bahwa modulo aritmatika modular bilangan prima membentuk [[medan (matematika)|medan]] atau [[medan berhingga]], sedangkan modulus lainnya hanya memberikan [[gelanggang (matematika)|gelanggang]] tetapi bukan sebuah medan.<ref>{{harvnb|Shahriari|2017}}, [https://books.google.com/books?id=GJwxDwAAQBAJ&pg=PA27 hal. 27–28].</ref>
▲on the result of the usual sum, difference, or product of integers.<ref>{{cite book|title=Algebra in Action: A Course in Groups, Rings, and Fields|volume=27|series=Pure and Applied Undergraduate Texts|first=Shahriar|last=Shahriari|author-link= Shahriar Shahriari |publisher=American Mathematical Society|year=2017|isbn=978-1-4704-2849-5|pages=20–21|url=https://books.google.com/books?id=GJwxDwAAQBAJ&pg=PA20}}</ref> Equality of integers corresponds to ''congruence'' in modular arithmetic:
<math>x</math> and <math>y</math> are congruent (written <math>x\equiv y</math> mod <math>n</math>) when they have the same remainder after division by <math>n</math>.<ref>{{harvnb|Dudley|1978}}, [https://books.google.com/books?id=tr7SzBTsk1UC&pg=PA28 Theorem 3, p. 28].</ref> However, in this system of numbers, [[Division (mathematics)|division]] by all nonzero numbers is possible if and only if the modulus is prime. For instance, with the prime number <math>7</math> as modulus, division by <math>3</math> is possible: <math>2/3\equiv 3\bmod{7}</math>, because [[clearing denominators]] by multiplying both sides by <math>3</math> gives the valid formula <math>2\equiv 9\bmod{7}</math>. However, with the composite modulus <math>6</math>, division by <math>3</math> is impossible. There is no valid solution to <math>2/3\equiv x\bmod{6}</math>: clearing denominators by multiplying by <math>3</math> causes the left-hand side to become <math>2</math> while the right-hand side becomes either <math>0</math> or <math>3</math>.
In the terminology of [[abstract algebra]], the ability to perform division means that modular arithmetic modulo a prime number forms a [[field (mathematics)|field]] or, more specifically, a [[finite field]], while other moduli only give a [[ring (mathematics)|ring]] but not a field.<ref>{{harvnb|Shahriari|2017}}, [https://books.google.com/books?id=GJwxDwAAQBAJ&pg=PA27 pp. 27–28].</ref>
Beberapa teorema tentang bilangan prima dirumuskan menggunakan aritmetika modular. Misalnya, [[teorema kecil Fermat]] menyatakan bahwa jika <math>a\not\equiv 0</math> (mod <math>p</math>), thenmaka <math>a^{p-1}\equiv 1</math> (mod <math>p</math>).<ref>{{harvnb|Ribenboim|2004}}, Fermat'sTeorema littlekecil theoremFermat anddan primitiveakar rootsprimitif modulo a primeprima, pphal. 17–21.</ref> ▼
Several theorems about primes can be formulated using modular arithmetic. For instance, [[Fermat's little theorem]] states that if
Menjumlahkan dari semua pilihan <math>a</math> diberikan persamaan
▲<math>a\not\equiv 0</math> (mod <math>p</math>), then <math>a^{p-1}\equiv 1</math> (mod <math>p</math>).<ref>{{harvnb|Ribenboim|2004}}, Fermat's little theorem and primitive roots modulo a prime, pp. 17–21.</ref>
Summing this over all choices of <math>a</math> gives the equation
:<math>\sum_{a=1}^{p-1} a^{p-1} \equiv (p-1) \cdot 1 \equiv -1 \pmod p,</math>
valid wheneverjika <math>p</math> isadalah primebilangan prima.
[[Giuga'sKonjektur conjectureGiuga]] saysmenyebutkan thatbahwa thispersamaan equationini isjuga alsomerupakan asyarat sufficientyang conditioncukup foruntuk <math>p</math> to bemenjadi primeprima.<ref>{{harvnb|Ribenboim|2004}}, The property of Giuga, pphal. 21–22.</ref>
[[Teorema Wilson's theorem]] saysmenyebutkan bahwa thatsebuah anbilangan integerbulat <math>p>1</math> isadalah primebilangan ifprima andjika onlydan ifhanya thejika [[factorialfaktorial]] <math>(p-1)!</math> iskongruen congruent todengan <math>-1</math> mod <math>p</math>. For a compositeUntuk {{nowrap|numberbilangan <math>\;n = r\cdot s\; </math>}} thisini cannottidak holdberlaku, sincekarena onesalah of its factorssatu dividesfaktornya bothmembagi {{mvar|n}} anddan <math>(n-1)!</math>, anddan sojadi <math>(n-1)!\equiv -1 \pmod{n}</math> isadalah impossiblehal mustahil.<ref>{{harvnb|Ribenboim|2004}}, The theorem of Wilson, phal. 21.</ref>
===''p''-adic numbers===
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