{ localUrl: '../page/unique_factorisation_domain.html', arbitalUrl: 'https://arbital.com/p/unique_factorisation_domain', rawJsonUrl: '../raw/5vk.json', likeableId: '3394', likeableType: 'page', myLikeValue: '0', likeCount: '3', dislikeCount: '0', likeScore: '3', individualLikes: [ 'EricBruylant', 'VladArber', 'JaimeSevillaMolina' ], pageId: 'unique_factorisation_domain', edit: '2', editSummary: '', prevEdit: '1', currentEdit: '2', wasPublished: 'true', type: 'wiki', title: 'Unique factorisation domain', clickbait: 'This is the correct way to abstract from the integers the fact that every integer can be written uniquely as a product of prime numbers.', textLength: '3891', alias: 'unique_factorisation_domain', externalUrl: '', sortChildrenBy: 'likes', hasVote: 'false', voteType: '', votesAnonymous: 'false', editCreatorId: 'PatrickStevens', editCreatedAt: '2016-08-14 14:08:00', pageCreatorId: 'PatrickStevens', pageCreatedAt: '2016-08-14 12:54:15', seeDomainId: '0', editDomainId: 'AlexeiAndreev', submitToDomainId: '0', isAutosave: 'false', isSnapshot: 'false', isLiveEdit: 'true', isMinorEdit: 'false', indirectTeacher: 'false', todoCount: '0', isEditorComment: 'false', isApprovedComment: 'true', isResolved: 'false', snapshotText: '', anchorContext: '', anchorText: '', anchorOffset: '0', mergedInto: '', isDeleted: 'false', viewCount: '28', text: '[summary(Technical): An [-5md] $R$ is said to be a *unique factorisation domain* if every nonzero non-[5mg unit] element of $R$ may be written as a product of [5m1 irreducibles], and moreover this product is unique up to reordering and multiplying by units.]\n\n[summary: A [3gq ring] $R$ is a *unique factorisation domain* if an analog of the [5rh] holds in it. The condition is as follows: every nonzero element, which does not have a multiplicative inverse, must be expressible as a product of [5m1 irreducibles], and the expression must be unique if we do not care about ordering or about multiplying by elements which have multiplicative inverses.]\n\n[3gq Ring theory] is the art of extracting properties from the [48l integers] and working out how they interact with each other.\nFrom this point of view, a *unique factorisation domain* is a ring in which the integers' [5rh] holds.\n\nThere have been various incorrect "proofs" of Fermat's Last Theorem throughout the ages; it turns out that if we assume the false "fact" that all subrings of [4zw $\\mathbb{C}$] are unique factorisation domains, then FLT is not particularly difficult to prove.\nThis is an example of where abstraction really is helpful: having a name for the concept of a UFD, and a few examples, makes it clear that it is not a trivial property and that it does need to be checked whenever we try and use it.\n\n# Formal statement\n\nLet $R$ be an [-5md].\nThen $R$ is a *unique factorisation domain* if every nonzero non-[5mg unit] element of $R$ may be expressed as a product of [5m1 irreducibles], and moreover this expression is unique up to reordering and multiplying by units.\n\n# Why ignore units?\n\nWe must set things up so that we don't care about units in the factorisations we discover.\nIndeed, if $u$ is a unit %%note:That is, it has a multiplicative inverse $u^{-1}$.%%, then $p \\times q$ is always equal to $(p \\times u) \\times (q \\times u^{-1})$, and this "looks like" a different factorisation into irreducibles.\n($p \\times u$ is irreducible if $p$ is irreducible and $u$ is a unit.)\nThe best we could possibly hope for is that the factorisation would be unique if we ignored multiplying by invertible elements, because those we may always forget about.\n\n## Example\n\nIn $\\mathbb{Z}$, the units are precisely $1$ and $-1$.\nWe have that $-10 = -1 \\times 5 \\times 2$ or $-5 \\times 2$ or $5 \\times -2$; we need these to be "the same" somehow.\n\nThe way we make them be "the same" is to insist that the $5$ and $-5$ are "the same" and the $2$ and $-2$ are "the same" (because they only differ by multiplication of the unit $-1$), and to note that $-1$ is not irreducible (because irreducibles are specifically defined to be non-unit) so $-1 \\times 5 \\times 2$ is not actually a factorisation into irreducibles.\n\nThat way, $-1 \\times 5 \\times 2$ is not a valid decomposition anyway, and $-5 \\times 2$ is just the same as $5 \\times -2$ because each of the irreducibles is the same up to multiplication by units.\n\n# Examples\n\n- Every [-5r5] is a unique factorisation domain. ([pid_implies_ufd Proof.]) This fact is not trivial! Therefore $\\mathbb{Z}$ is a UFD, though we can also prove this directly; this is the [5rh].\n- $\\mathbb{Z}[-\\sqrt{3}]$ is *not* a UFD. Indeed, $4 = 2 \\times 2$ but also $(1+\\sqrt{-3})(1-\\sqrt{-3})$; these are both decompositions into irreducible elements. (See the page on [5m1 irreducibles] for a proof that $2$ is irreducible; the same proof can be adapted to show that $1 \\pm \\sqrt{-3}$ are both irreducible.)\n\n# Properties\n\n- If it is hard to test for uniqueness up to reordering and multiplying by units, there is an easier but equivalent condition to check: an integral domain is a unique factorisation domain if and only if every element can be written (not necessarily uniquely) as a product of irreducibles, and all irreducibles are [5m2 prime]. ([alternative_condition_for_ufd Proof.])', metaText: '', isTextLoaded: 'true', isSubscribedToDiscussion: 'false', isSubscribedToUser: 'false', isSubscribedAsMaintainer: 'false', discussionSubscriberCount: '1', maintainerCount: '1', userSubscriberCount: '0', lastVisit: '', hasDraft: 'false', votes: [], voteSummary: 'null', muVoteSummary: '0', voteScaling: '0', currentUserVote: '-2', voteCount: '0', lockedVoteType: '', maxEditEver: '0', redLinkCount: '0', lockedBy: '', lockedUntil: '', nextPageId: '', prevPageId: '', usedAsMastery: 'false', proposalEditNum: '0', permissions: { edit: { has: 'false', reason: 'You don't have domain permission to edit this page' }, proposeEdit: { has: 'true', reason: '' }, delete: { has: 'false', reason: 'You don't have domain permission to delete this page' }, comment: { has: 'false', reason: 'You can't comment in this domain because you are not a member' }, proposeComment: { has: 'true', reason: '' } }, summaries: {}, creatorIds: [ 'PatrickStevens' ], childIds: [], parentIds: [ 'algebraic_ring' ], commentIds: [], questionIds: [], tagIds: [], relatedIds: [], markIds: [], explanations: [], learnMore: [], requirements: [], subjects: [], lenses: [], lensParentId: '', pathPages: [], learnMoreTaughtMap: {}, learnMoreCoveredMap: {}, learnMoreRequiredMap: {}, editHistory: {}, domainSubmissions: {}, answers: [], answerCount: '0', commentCount: '0', newCommentCount: '0', linkedMarkCount: '0', changeLogs: [ { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '18730', pageId: 'unique_factorisation_domain', userId: 'PatrickStevens', edit: '2', type: 'newEdit', createdAt: '2016-08-14 14:08:00', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '18729', pageId: 'unique_factorisation_domain', userId: 'PatrickStevens', edit: '0', type: 'newParent', createdAt: '2016-08-14 12:54:43', auxPageId: 'algebraic_ring', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '18727', pageId: 'unique_factorisation_domain', userId: 'PatrickStevens', edit: '1', type: 'newEdit', createdAt: '2016-08-14 12:54:15', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' } ], feedSubmissions: [], searchStrings: {}, hasChildren: 'false', hasParents: 'true', redAliases: {}, improvementTagIds: [], nonMetaTagIds: [], todos: [], slowDownMap: 'null', speedUpMap: 'null', arcPageIds: 'null', contentRequests: {} }