{ localUrl: '../page/field_homomorphism_is_trivial_or_injective.html', arbitalUrl: 'https://arbital.com/p/field_homomorphism_is_trivial_or_injective', rawJsonUrl: '../raw/76h.json', likeableId: '0', likeableType: 'page', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], pageId: 'field_homomorphism_is_trivial_or_injective', edit: '1', editSummary: '', prevEdit: '0', currentEdit: '1', wasPublished: 'true', type: 'wiki', title: 'Field homomorphism is trivial or injective', clickbait: 'Field homomorphisms preserve a *lot* of structure; they preserve so much structure that they are always either injective or totally boring.', textLength: '1961', alias: 'field_homomorphism_is_trivial_or_injective', externalUrl: '', sortChildrenBy: 'likes', hasVote: 'false', voteType: '', votesAnonymous: 'false', editCreatorId: 'PatrickStevens', editCreatedAt: '2016-12-31 14:23:18', pageCreatorId: 'PatrickStevens', pageCreatedAt: '2016-12-31 14:23:18', seeDomainId: '0', editDomainId: '163', submitToDomainId: '0', isAutosave: 'false', isSnapshot: 'false', isLiveEdit: 'true', isMinorEdit: 'false', indirectTeacher: 'false', todoCount: '0', isEditorComment: 'false', isApprovedComment: 'false', isResolved: 'false', snapshotText: '', anchorContext: '', anchorText: '', anchorOffset: '0', mergedInto: '', isDeleted: 'false', viewCount: '19', text: '[summary(Technical): Let $F$ and $G$ be [481 fields], and let $f: F \\to G$ be a [-field_homomorphism]. Then $f$ either is the constant $0$ map, or it is [4b7 injective].]\n\n[summary: A structure-preserving [3jy map] between two [481 fields] turns out either to be totally trivial (sending every element to $0$) or it preserves so much structure that its [3lh image] is an embedded copy of the [3js domain]. More succinctly, if $f$ is a field homomorphism, then $f$ is either the constant $0$ map, or it is [4b7 injective].]\n\nLet $F$ and $G$ be [481 fields], and let $f: F \\to G$ be a [-field_homomorphism]. Then one of the following is the case:\n\n - $f$ is the constant $0$ map: for every $x \\in F$, we have $f(x) = 0_G$.\n - $f$ is [4b7 injective].\n\n# Proof\n\nLet $f: F \\to G$ be non-constant.\nWe need to show that $f$ is injective; equivalently, for any pair $x,y$ of elements with $f(x) = f(y)$, we need to show that $x = y$.\n\nSuppose $f(x) = f(y)$.\nThen we have $f(x)-f(y) = 0_G$; so $f(x-y) = 0_G$ because $f$ is a field homomorphism and so respects the "subtraction" operation.\nHence in fact it is enough to show the following sub-result:\n\n> Suppose $f$ is non-constant. If $f(z) = 0_G$, then $z = 0_F$.\n\nOnce we have done this, we simply let $z = x-y$.\n\n## Proof of sub-result\n\nSuppose $f(z) = 0_G$ but that $z$ is not $0_F$, so we may find its multiplicative inverse $z^{-1}$.\n\nThen $f(z^{-1}) f(z) = f(z^{-1}) \\times 0_G = 0_G$; but $f$ is a homomorphism, so $f(z^{-1} \\times z) = 0_G$, and so $f(1_F) = 0_G$.\n\nBut this contradicts that the [-49z], because we may consider $f$ to be a [-47t] between the *multiplicative groups* $F \\setminus \\{ 0_F \\}$ and $G \\setminus \\{0_G\\}$, whereupon $1_F$ is the identity of $F \\setminus \\{0_F\\}$, and $1_G$ is the identity of $F \\setminus \\{0_G\\}$.\n\nOur assumption on $z$ was that $z \\not = 0_F$, so the contradiction means that if $f(z) = 0_G$ then $z = 0_F$.\nThis proves the sub-result and hence the main theorem.', 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_field' ], commentIds: [], questionIds: [], tagIds: [ 'proof_meta_tag' ], 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: '3909', likeableType: 'changeLog', myLikeValue: '0', likeCount: '1', dislikeCount: '0', likeScore: '1', individualLikes: [], id: '21196', pageId: 'field_homomorphism_is_trivial_or_injective', userId: 'PatrickStevens', edit: '1', type: 'newEdit', createdAt: '2016-12-31 14:23:18', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' } ], feedSubmissions: [], searchStrings: {}, hasChildren: 'false', hasParents: 'true', redAliases: {}, improvementTagIds: [], nonMetaTagIds: [], todos: [], slowDownMap: 'null', speedUpMap: 'null', arcPageIds: 'null', contentRequests: {} }