{"history":[],"article":{"id":"113ef9ab-7457-4c63-828e-c9b3fe19c6cf","title":"Fundamental theorem of arithmetic","slug":"fundamental-theorem-of-arithmetic","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), the **fundamental theorem of arithmetic**, also called the **unique factorization theorem** and **prime factorization theorem**, states that every [integer](https://bliptext.com/articles/integer) greater than 1 can be represented uniquely as a product of [prime numbers](https://bliptext.com/articles/prime-number), [up to](https://bliptext.com/articles/up-to) the order of the factors.\nFor example,\nThe theorem says two things about this example: first, that 1200 be represented as a product of primes, and second, that no matter how this is done, there will always be exactly four 2s, one 3, two 5s, and no other primes in the product.\nThe requirement that the factors be prime is necessary: factorizations containing [composite numbers](https://bliptext.com/articles/composite-number) may not be unique\n(for example, ).\nThis theorem is one of the main [reasons why 1 is not considered a prime number](https://bliptext.com/articles/prime-number): if 1 were prime, then factorization into primes would not be unique; for example,\nThe theorem generalizes to other [algebraic structures](https://bliptext.com/articles/algebraic-structure) that are called [unique factorization domains](https://bliptext.com/articles/unique-factorization-domain) and include [principal ideal domains](https://bliptext.com/articles/principal-ideal-domain), [Euclidean domains](https://bliptext.com/articles/euclidean-domain), and [polynomial rings](https://bliptext.com/articles/polynomial-ring) over a [field](https://bliptext.com/articles/field-mathematics).\nHowever, the theorem does not hold for [algebraic integers](https://bliptext.com/articles/algebraic-integer).\nThis failure of unique factorization is one of the reasons for the difficulty of the proof of [Fermat's Last Theorem](https://bliptext.com/articles/fermat-s-last-theorem).\nThe implicit use of unique factorization in rings of algebraic integers is behind the error of many of the numerous false proofs that have been written during the 358 years between [Fermat's](https://bliptext.com/articles/pierre-de-fermat) statement and [Wiles's proof](https://bliptext.com/articles/wiles-s-proof-of-fermat-s-last-theorem).\n# [In Disquisitiones Arithmeticae (1801) Gauss proved the unique factorization theorem and used it to prove the law of quadratic reciprocity. | upload.wikimedia.org/wikipedia/commons/e/e3/Disqvisitiones-800.jpg]\n\n# History\nThe fundamental theorem can be derived from Book VII, propositions 30, 31 and 32, and Book IX, proposition 14 of [Euclid](https://bliptext.com/articles/euclid)'s [Elements](https://bliptext.com/articles/euclid-s-elements).\n(In modern terminology: if a prime p divides the product ab, then p divides either a or b or both.) Proposition 30 is referred to as [Euclid's lemma](https://bliptext.com/articles/euclid-s-lemma), and it is the key in the proof of the fundamental theorem of arithmetic.\n(In modern terminology: every integer greater than one is divided evenly by some prime number.) Proposition 31 is proved directly by [infinite descent](https://bliptext.com/articles/proof-by-infinite-descent).\nProposition 32 is derived from proposition 31, and proves that the decomposition is possible.\n(In modern terminology: a [least common multiple](https://bliptext.com/articles/least-common-multiple) of several prime numbers is not a multiple of any other prime number.) Book IX, proposition 14 is derived from Book VII, proposition 30, and proves partially that the decomposition is unique – a point critically noted by [André Weil](https://bliptext.com/articles/andr-weil).\nIndeed, in this proposition the exponents are all equal to one, so nothing is said for the general case.\nWhile [Euclid](https://bliptext.com/articles/euclid) took the first step on the way to the existence of prime factorization, [Kamāl al-Dīn al-Fārisī](https://bliptext.com/articles/kam-l-al-d-n-al-f-ris) took the final step and stated for the first time the fundamental theorem of arithmetic.\nArticle 16 of [Gauss](https://bliptext.com/articles/carl-friedrich-gauss)'s [Disquisitiones Arithmeticae](https://bliptext.com/articles/disquisitiones-arithmeticae) is an early modern statement and proof employing [modular arithmetic](https://bliptext.com/articles/modular-arithmetic).\n\n# Applications\n\n## Canonical representation of a positive integer\nEvery positive integer\nn > 1\ncan be represented in exactly one way as a product of prime powers\nwhere\np1 < p2 < ...\n< pk\nare primes and the\nni\nare positive integers.\nThis representation is commonly extended to all positive integers, including 1, by the convention that the [empty product](https://bliptext.com/articles/empty-product) is equal to 1 (the [empty product](https://bliptext.com/articles/empty-product) corresponds to\nk = 0\n).\nThis representation is called the **canonical representation** of\nn\n, or the **standard form** of n.\nFor example,\nFactors\np0 = 1\nmay be inserted without changing the value of\nn\n(for example,\n1000 = 23×30×53\n). In fact, any positive integer can be uniquely represented as an [infinite product](https://bliptext.com/articles/infinite-product) taken over all the positive prime numbers, as where a finite number of the\nni\nare positive integers, and the others are zero.\nAllowing negative exponents provides a canonical form for positive [rational numbers](https://bliptext.com/articles/rational-number).\n\n## Arithmetic operations\nThe canonical representations of the product, [greatest common divisor](https://bliptext.com/articles/greatest-common-divisor) (GCD), and [least common multiple](https://bliptext.com/articles/least-common-multiple) (LCM) of two numbers a and b can be expressed simply in terms of the canonical representations of a and b themselves:\nHowever, [integer factorization](https://bliptext.com/articles/integer-factorization), especially of large numbers, is much more difficult than computing products, GCDs, or LCMs.\nSo these formulas have limited use in practice.\n\n## Arithmetic functions\nMany arithmetic functions are defined using the canonical representation.\nIn particular, the values of [additive](https://bliptext.com/articles/additive-function) and [multiplicative](https://bliptext.com/articles/multiplicative-function) functions are determined by their values on the powers of prime numbers.\n\n# Proof\nThe proof uses [Euclid's lemma](https://bliptext.com/articles/euclid-s-lemma) (Elements VII, 30): If a prime [divides](https://bliptext.com/articles/divisor) the product of two integers, then it must divide at least one of these integers.\n\n## Existence\nIt must be shown that every integer greater than\n1\nis either prime or a product of primes.\nFirst,\n2\nis prime.\nThen, by [strong induction](https://bliptext.com/articles/strong-induction), assume this is true for all numbers greater than\n1\nand less than\nn\n. If\nn\nis prime, there is nothing more to prove.\nOtherwise, there are integers\na\nand\nb\n, where\nn = a b\n, and\n1 < a ≤ b < n\n. By the induction hypothesis,\na = p1 p2 ⋅⋅⋅ pj\nand\nb = q1 q2 ⋅⋅⋅ qk\nare products of primes.\nBut then\nn = a b = p1 p2 ⋅⋅⋅ pj q1 q2 ⋅⋅⋅ qk\nis a product of primes.\n\n## Uniqueness\nSuppose, to the contrary, there is an integer that has two distinct prime factorizations.\nLet\nn\nbe the least such integer and write\nn = p1 p2 ... pj = q1 q2 ...\nqk\n, where each\npi\nand\nqi\nis prime.\nWe see that\np1\ndivides\nq1 q2 ...\nqk\n, so\np1\ndivides some\nqi\nby [Euclid's lemma](https://bliptext.com/articles/euclid-s-lemma).\nWithout loss of generality, say\np1\ndivides\nq1\n. Since\np1\nand\nq1\nare both prime, it follows that\np1 = q1\n. Returning to our factorizations of\nn\n, we may cancel these two factors to conclude that\np2 ... pj = q2 ...\nqk\n. We now have two distinct prime factorizations of some integer strictly smaller than\nn\n, which contradicts the minimality of\nn\n.\n\n## Uniqueness without Euclid's lemma\nThe fundamental theorem of arithmetic can also be proved without using Euclid's lemma.\nThe proof that follows is inspired by Euclid's original version of the [Euclidean algorithm](https://bliptext.com/articles/euclidean-algorithm).\nAssume that s is the smallest positive integer which is the product of prime numbers in two different ways.\nIncidentally, this implies that s, if it exists, must be a [composite number](https://bliptext.com/articles/composite-number) greater than 1.\nNow, say\nEvery p_i must be distinct from every q_j.\nOtherwise, if say p_i=q_j, then there would exist some positive integer that is smaller than s and has two distinct prime factorizations.\nOne may also suppose that p_1 < q_1, by exchanging the two factorizations, if needed.\nSetting and one has\nAlso, since p_1 < q_1, one has Q < P. It then follows that\nAs the positive integers less than s have been supposed to have a unique prime factorization, p_1 must occur in the factorization of either q_1-p_1 or Q. The latter case is impossible, as Q, being smaller than s, must have a unique prime factorization, and p_1 differs from every q_j.\nThe former case is also impossible, as, if p_1 is a divisor of q_1-p_1, it must be also a divisor of q_1, which is impossible as p_1 and q_1 are distinct primes.\nTherefore, there cannot exist a smallest integer with more than a single distinct prime factorization.\nEvery positive integer must either be a prime number itself, which would factor uniquely, or a composite that also factors uniquely into primes, or in the case of the integer 1, not factor into any prime.\n\n# Generalizations\nThe first generalization of the theorem is found in Gauss's second monograph (1832) on [biquadratic reciprocity](https://bliptext.com/articles/biquadratic-reciprocity).\nThis paper introduced what is now called the [ring](https://bliptext.com/articles/ring-theory) of [Gaussian integers](https://bliptext.com/articles/gaussian-integer), the set of all [complex numbers](https://bliptext.com/articles/complex-number) a + bi where a and b are integers.\nIt is now denoted by He showed that this ring has the four units ±1 and ±i, that the non-zero, non-unit numbers fall into two classes, primes and composites, and that (except for order), the composites have unique factorization as a product of primes ([up to](https://bliptext.com/articles/up-to) the order and multiplication by units).\nSimilarly, in 1844 while working on [cubic reciprocity](https://bliptext.com/articles/cubic-reciprocity), [Eisenstein](https://bliptext.com/articles/gotthold-eisenstein) introduced the ring, where is a cube [root of unity](https://bliptext.com/articles/root-of-unity).\nThis is the ring of [Eisenstein integers](https://bliptext.com/articles/eisenstein-integer), and he proved it has the six units and that it has unique factorization.\nHowever, it was also discovered that unique factorization does not always hold.\nAn example is given by.\nIn this ring one has\nExamples like this caused the notion of \"prime\" to be modified.\nIn it can be proven that if any of the factors above can be represented as a product, for example, 2 = ab, then one of a or b must be a unit.\nThis is the traditional definition of \"prime\".\nIt can also be proven that none of these factors obeys Euclid's lemma; for example, 2 divides neither (1 + √−5) nor (1 − √−5) even though it divides their product 6.\nIn [algebraic number theory](https://bliptext.com/articles/algebraic-number-theory) 2 is called [irreducible](https://bliptext.com/articles/irreducible-element) in (only divisible by itself or a unit) but not [prime](https://bliptext.com/articles/prime-element) in (if it divides a product it must divide one of the factors).\nThe mention of is required because 2 is prime and irreducible in \\mathbb{Z}.\nUsing these definitions it can be proven that in any [integral domain](https://bliptext.com/articles/integral-domain) a prime must be irreducible.\nEuclid's classical lemma can be rephrased as \"in the ring of integers \\mathbb{Z} every irreducible is prime\".\nThis is also true in and but not in\nThe rings in which factorization into irreducibles is essentially unique are called [unique factorization domains](https://bliptext.com/articles/unique-factorization-domain).\nImportant examples are [polynomial rings](https://bliptext.com/articles/polynomial-ring) over the integers or over a [field](https://bliptext.com/articles/field-mathematics), [Euclidean domains](https://bliptext.com/articles/euclidean-domain) and [principal ideal domains](https://bliptext.com/articles/principal-ideal-domain).\nIn 1843 [Kummer](https://bliptext.com/articles/ernst-kummer) introduced the concept of [ideal number](https://bliptext.com/articles/ideal-number), which was developed further by [Dedekind](https://bliptext.com/articles/richard-dedekind) (1876) into the modern theory of [ideals](https://bliptext.com/articles/ideal-ring-theory), special subsets of rings.\nMultiplication is defined for ideals, and the rings in which they have unique factorization are called [Dedekind domains](https://bliptext.com/articles/dedekind-domain).\nThere is a version of [unique factorization for ordinals](https://bliptext.com/articles/ordinal-arithmetic), though it requires some additional conditions to ensure uniqueness.\nAny commutative Möbius monoid satisfies a unique factorization theorem and thus possesses arithmetical properties similar to those of the multiplicative semigroup of positive integers.\nFundamental Theorem of Arithmetic is, in fact, a special case of the unique factorization theorem in commutative Möbius monoids.","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T17:21:00.317Z","updated_at":"2025-01-25T17:21:00.317Z","search_vector":"'arithmet':4 'fundament':1 'theorem':2","search_content":" \n \n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), the **fundamental theorem of arithmetic**, also called the **unique factorization theorem** and **prime factorization theorem**, states that every [integer](https://bliptext.com/articles/integer) greater than 1 can be represented uniquely as a product of [prime numbers](https://bliptext.com/articles/prime-number), [up to](https://bliptext.com/articles/up-to) the order of the factors.\nFor example,\nThe theorem says two things about this example: first, that 1200 be represented as a product of primes, and second, that no matter how this is done, there will always be exactly four 2s, one 3, two 5s, and no other primes in the product.\nThe requirement that the factors be prime is necessary: factorizations containing [composite numbers](https://bliptext.com/articles/composite-number) may not be unique\n(for example, ).\nThis theorem is one of the main [reasons why 1 is not considered a prime number](https://bliptext.com/articles/prime-number): if 1 were prime, then factorization into primes would not be unique; for example,\nThe theorem generalizes to other [algebraic structures](https://bliptext.com/articles/algebraic-structure) that are called [unique factorization domains](https://bliptext.com/articles/unique-factorization-domain) and include [principal ideal domains](https://bliptext.com/articles/principal-ideal-domain), [Euclidean domains](https://bliptext.com/articles/euclidean-domain), and [polynomial rings](https://bliptext.com/articles/polynomial-ring) over a [field](https://bliptext.com/articles/field-mathematics).\nHowever, the theorem does not hold for [algebraic integers](https://bliptext.com/articles/algebraic-integer).\nThis failure of unique factorization is one of the reasons for the difficulty of the proof of [Fermat's Last Theorem](https://bliptext.com/articles/fermat-s-last-theorem).\nThe implicit use of unique factorization in rings of algebraic integers is behind the error of many of the numerous false proofs that have been written during the 358 years between [Fermat's](https://bliptext.com/articles/pierre-de-fermat) statement and [Wiles's proof](https://bliptext.com/articles/wiles-s-proof-of-fermat-s-last-theorem).\n# [In Disquisitiones Arithmeticae (1801) Gauss proved the unique factorization theorem and used it to prove the law of quadratic reciprocity. | upload.wikimedia.org/wikipedia/commons/e/e3/Disqvisitiones-800.jpg]\n\n# History\nThe fundamental theorem can be derived from Book VII, propositions 30, 31 and 32, and Book IX, proposition 14 of [Euclid](https://bliptext.com/articles/euclid)'s [Elements](https://bliptext.com/articles/euclid-s-elements).\n(In modern terminology: if a prime p divides the product ab, then p divides either a or b or both.) Proposition 30 is referred to as [Euclid's lemma](https://bliptext.com/articles/euclid-s-lemma), and it is the key in the proof of the fundamental theorem of arithmetic.\n(In modern terminology: every integer greater than one is divided evenly by some prime number.) Proposition 31 is proved directly by [infinite descent](https://bliptext.com/articles/proof-by-infinite-descent).\nProposition 32 is derived from proposition 31, and proves that the decomposition is possible.\n(In modern terminology: a [least common multiple](https://bliptext.com/articles/least-common-multiple) of several prime numbers is not a multiple of any other prime number.) Book IX, proposition 14 is derived from Book VII, proposition 30, and proves partially that the decomposition is unique – a point critically noted by [André Weil](https://bliptext.com/articles/andr-weil).\nIndeed, in this proposition the exponents are all equal to one, so nothing is said for the general case.\nWhile [Euclid](https://bliptext.com/articles/euclid) took the first step on the way to the existence of prime factorization, [Kamāl al-Dīn al-Fārisī](https://bliptext.com/articles/kam-l-al-d-n-al-f-ris) took the final step and stated for the first time the fundamental theorem of arithmetic.\nArticle 16 of [Gauss](https://bliptext.com/articles/carl-friedrich-gauss)'s [Disquisitiones Arithmeticae](https://bliptext.com/articles/disquisitiones-arithmeticae) is an early modern statement and proof employing [modular arithmetic](https://bliptext.com/articles/modular-arithmetic).\n\n# Applications\n\n## Canonical representation of a positive integer\nEvery positive integer\nn > 1\ncan be represented in exactly one way as a product of prime powers\nwhere\np1 < p2 < ...\n< pk\nare primes and the\nni\nare positive integers.\nThis representation is commonly extended to all positive integers, including 1, by the convention that the [empty product](https://bliptext.com/articles/empty-product) is equal to 1 (the [empty product](https://bliptext.com/articles/empty-product) corresponds to\nk = 0\n).\nThis representation is called the **canonical representation** of\nn\n, or the **standard form** of n.\nFor example,\nFactors\np0 = 1\nmay be inserted without changing the value of\nn\n(for example,\n1000 = 23×30×53\n). In fact, any positive integer can be uniquely represented as an [infinite product](https://bliptext.com/articles/infinite-product) taken over all the positive prime numbers, as where a finite number of the\nni\nare positive integers, and the others are zero.\nAllowing negative exponents provides a canonical form for positive [rational numbers](https://bliptext.com/articles/rational-number).\n\n## Arithmetic operations\nThe canonical representations of the product, [greatest common divisor](https://bliptext.com/articles/greatest-common-divisor) (GCD), and [least common multiple](https://bliptext.com/articles/least-common-multiple) (LCM) of two numbers a and b can be expressed simply in terms of the canonical representations of a and b themselves:\nHowever, [integer factorization](https://bliptext.com/articles/integer-factorization), especially of large numbers, is much more difficult than computing products, GCDs, or LCMs.\nSo these formulas have limited use in practice.\n\n## Arithmetic functions\nMany arithmetic functions are defined using the canonical representation.\nIn particular, the values of [additive](https://bliptext.com/articles/additive-function) and [multiplicative](https://bliptext.com/articles/multiplicative-function) functions are determined by their values on the powers of prime numbers.\n\n# Proof\nThe proof uses [Euclid's lemma](https://bliptext.com/articles/euclid-s-lemma) (Elements VII, 30): If a prime [divides](https://bliptext.com/articles/divisor) the product of two integers, then it must divide at least one of these integers.\n\n## Existence\nIt must be shown that every integer greater than\n1\nis either prime or a product of primes.\nFirst,\n2\nis prime.\nThen, by [strong induction](https://bliptext.com/articles/strong-induction), assume this is true for all numbers greater than\n1\nand less than\nn\n. If\nn\nis prime, there is nothing more to prove.\nOtherwise, there are integers\na\nand\nb\n, where\nn = a b\n, and\n1 < a ≤ b < n\n. By the induction hypothesis,\na = p1 p2 ⋅⋅⋅ pj\nand\nb = q1 q2 ⋅⋅⋅ qk\nare products of primes.\nBut then\nn = a b = p1 p2 ⋅⋅⋅ pj q1 q2 ⋅⋅⋅ qk\nis a product of primes.\n\n## Uniqueness\nSuppose, to the contrary, there is an integer that has two distinct prime factorizations.\nLet\nn\nbe the least such integer and write\nn = p1 p2 ... pj = q1 q2 ...\nqk\n, where each\npi\nand\nqi\nis prime.\nWe see that\np1\ndivides\nq1 q2 ...\nqk\n, so\np1\ndivides some\nqi\nby [Euclid's lemma](https://bliptext.com/articles/euclid-s-lemma).\nWithout loss of generality, say\np1\ndivides\nq1\n. Since\np1\nand\nq1\nare both prime, it follows that\np1 = q1\n. Returning to our factorizations of\nn\n, we may cancel these two factors to conclude that\np2 ... pj = q2 ...\nqk\n. We now have two distinct prime factorizations of some integer strictly smaller than\nn\n, which contradicts the minimality of\nn\n.\n\n## Uniqueness without Euclid's lemma\nThe fundamental theorem of arithmetic can also be proved without using Euclid's lemma.\nThe proof that follows is inspired by Euclid's original version of the [Euclidean algorithm](https://bliptext.com/articles/euclidean-algorithm).\nAssume that s is the smallest positive integer which is the product of prime numbers in two different ways.\nIncidentally, this implies that s, if it exists, must be a [composite number](https://bliptext.com/articles/composite-number) greater than 1.\nNow, say\nEvery p_i must be distinct from every q_j.\nOtherwise, if say p_i=q_j, then there would exist some positive integer that is smaller than s and has two distinct prime factorizations.\nOne may also suppose that p_1 < q_1, by exchanging the two factorizations, if needed.\nSetting and one has\nAlso, since p_1 < q_1, one has Q < P. It then follows that\nAs the positive integers less than s have been supposed to have a unique prime factorization, p_1 must occur in the factorization of either q_1-p_1 or Q. The latter case is impossible, as Q, being smaller than s, must have a unique prime factorization, and p_1 differs from every q_j.\nThe former case is also impossible, as, if p_1 is a divisor of q_1-p_1, it must be also a divisor of q_1, which is impossible as p_1 and q_1 are distinct primes.\nTherefore, there cannot exist a smallest integer with more than a single distinct prime factorization.\nEvery positive integer must either be a prime number itself, which would factor uniquely, or a composite that also factors uniquely into primes, or in the case of the integer 1, not factor into any prime.\n\n# Generalizations\nThe first generalization of the theorem is found in Gauss's second monograph (1832) on [biquadratic reciprocity](https://bliptext.com/articles/biquadratic-reciprocity).\nThis paper introduced what is now called the [ring](https://bliptext.com/articles/ring-theory) of [Gaussian integers](https://bliptext.com/articles/gaussian-integer), the set of all [complex numbers](https://bliptext.com/articles/complex-number) a + bi where a and b are integers.\nIt is now denoted by He showed that this ring has the four units ±1 and ±i, that the non-zero, non-unit numbers fall into two classes, primes and composites, and that (except for order), the composites have unique factorization as a product of primes ([up to](https://bliptext.com/articles/up-to) the order and multiplication by units).\nSimilarly, in 1844 while working on [cubic reciprocity](https://bliptext.com/articles/cubic-reciprocity), [Eisenstein](https://bliptext.com/articles/gotthold-eisenstein) introduced the ring, where is a cube [root of unity](https://bliptext.com/articles/root-of-unity).\nThis is the ring of [Eisenstein integers](https://bliptext.com/articles/eisenstein-integer), and he proved it has the six units and that it has unique factorization.\nHowever, it was also discovered that unique factorization does not always hold.\nAn example is given by.\nIn this ring one has\nExamples like this caused the notion of \"prime\" to be modified.\nIn it can be proven that if any of the factors above can be represented as a product, for example, 2 = ab, then one of a or b must be a unit.\nThis is the traditional definition of \"prime\".\nIt can also be proven that none of these factors obeys Euclid's lemma; for example, 2 divides neither (1 + √−5) nor (1 − √−5) even though it divides their product 6.\nIn [algebraic number theory](https://bliptext.com/articles/algebraic-number-theory) 2 is called [irreducible](https://bliptext.com/articles/irreducible-element) in (only divisible by itself or a unit) but not [prime](https://bliptext.com/articles/prime-element) in (if it divides a product it must divide one of the factors).\nThe mention of is required because 2 is prime and irreducible in \\mathbb{Z}.\nUsing these definitions it can be proven that in any [integral domain](https://bliptext.com/articles/integral-domain) a prime must be irreducible.\nEuclid's classical lemma can be rephrased as \"in the ring of integers \\mathbb{Z} every irreducible is prime\".\nThis is also true in and but not in\nThe rings in which factorization into irreducibles is essentially unique are called [unique factorization domains](https://bliptext.com/articles/unique-factorization-domain).\nImportant examples are [polynomial rings](https://bliptext.com/articles/polynomial-ring) over the integers or over a [field](https://bliptext.com/articles/field-mathematics), [Euclidean domains](https://bliptext.com/articles/euclidean-domain) and [principal ideal domains](https://bliptext.com/articles/principal-ideal-domain).\nIn 1843 [Kummer](https://bliptext.com/articles/ernst-kummer) introduced the concept of [ideal number](https://bliptext.com/articles/ideal-number), which was developed further by [Dedekind](https://bliptext.com/articles/richard-dedekind) (1876) into the modern theory of [ideals](https://bliptext.com/articles/ideal-ring-theory), special subsets of rings.\nMultiplication is defined for ideals, and the rings in which they have unique factorization are called [Dedekind domains](https://bliptext.com/articles/dedekind-domain).\nThere is a version of [unique factorization for ordinals](https://bliptext.com/articles/ordinal-arithmetic), though it requires some additional conditions to ensure uniqueness.\nAny commutative Möbius monoid satisfies a unique factorization theorem and thus possesses arithmetical properties similar to those of the multiplicative semigroup of positive integers.\nFundamental Theorem of Arithmetic is, in fact, a special case of the unique factorization theorem in commutative Möbius monoids.","revisionCount":0},"hasMore":false,"nextCursor":null}