Group

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• The integers are a Euclidean domain 000JFQGDXG
• The characteristic of an integral domain is zero or prime 00758QM6JJ
• Zero ring 008TBCYDPX
• Principal ideal 00A2EDFT9Q
• Laws of exponents in a group 00DRK90Z6K
• Root of a polynomial 00EHX6FWC0
• Orbit 00F2JFA41J
• Transcendental element 00SW8B91E7
• Irreducible polynomial 00X2A9G9W3
• A finite group is nilpotent iff it is the product of its Sylow subgroups 00Y9N80VSS
• Uniqueness of finite fields of a given order 010PKCQPXP
• Image of a homomorphism 0112DTS464
• Evaluation homomorphism 0125QVNCFQ
• Existence of splitting fields 01441KDGBY
• Semidirect product 014E1DZQE8
• Cayley's theorem 0151VYVX27
• Unit of a ring 019NFGFPR5
• Characteristic subgroup 01CS8Q2APC
• Normality via left and right cosets 01E7SWYJX3
• Extensions of solvable groups are solvable 01EY0EM7RP
• Uniqueness of the algebraic closure 01VQ7C4FTB
• Free group 020Z5SEN27
• Euclidean algorithm in a Euclidean domain 023VY493K0
• Quotients of solvable groups are solvable 027ZQS04SJ
• Infinite cyclic group 02B1T50JD1
• Eisenstein's criterion 02BPKD7TMD
• Transposition 02DTRYKGE8
• Existence of maximal ideals 02HJYT53F7
• Intermediate field 02NBKVQFRQ
• Splitting field 02NZZ1J0FH
• Algebraically closed field 02SJG1HC32
• Ideal generated by a subset 030TZ0DQWT
• Order of a cyclic subgroup equals the order of its generator 031M8HX7DA
• Freshman's dream 0396VGFA78
• Norm on the Gaussian integers 03AN1V4ZFJ
• Separable polynomial 03J8BS7FMV
• Multiplicity of a root 03XVP8Q4GY
• Subgroups of a finite cyclic group correspond to divisors of its order 0423TXR9AN
• Order of a group is the index times the order of the subgroup 04DC0SRN2K
• Circle group 04EAFVFTF7
• Divisibility in a commutative ring 04GNK7K4VP
• Index of a subgroup 04QEBF8S49
• Cyclic groups are abelian 04RG02S6BA
• Galois field 04SGPNT8R9
• Irreducible element 052SDEPZH7
• Associate elements 053E86NWVK
• The dihedral group of order 8 054SBCQ93T
• PID implies UFD 055HT85T3S
• Stabilisers of points in the same orbit are conjugate 05BM304NRX
• Left coset 05FD317V2W
• Inner automorphism group is isomorphic to G modulo its centre 05HYGWFG63
• The rationals as the field of fractions of the integers 05K3J8J8T2
• Groups of prime order are cyclic 05MM4EK4SA
• Subgroup criterion 05N4TQKPH0
• Group of units of a ring 05THZKR9AN
• Maximal ideals are prime 05WAF5T0GZ
• Subgroups of solvable groups are solvable 05Y0F9M0TS
• Right coset 060Z37VVB6
• Repeated roots and the derivative 067F8YVYMH
• Uniqueness of splitting fields 069A3BC3EP
• Symmetric polynomial 06CTFY2DEZ
• P-group 06DAH3VS6M
• Z[x] is not a principal ideal domain 06E3AG9TM2
• Prime ideal 06HC6HK0Z8
• Classification of finite simple groups 06J6P03MG7
• The minimal polynomial divides every polynomial vanishing at the element 06K7SARZKR
• Commutator of two elements 06RGT4V6Y6
• Units of the Gaussian integers 06RH4XAGRY
• Unique factorisation domain 06YMXJMXYS
• Abelian group 071DDFVG4J
• Cyclic groups of the same order are isomorphic 0736AXH2ME
• Third isomorphism theorem for rings 073C80EC63
• Even and odd permutations 0771AJP4BX
• Classification of finite abelian groups 07CQR9BW3R
• Disjoint cycles commute 07E6D8R4D6
• Subfield 07QY5RPFBR
• Rational root theorem 0819CQ1V8H
• Finite p-groups are nilpotent 081WTKYX6C
• Subring 0859D58WPD
• Group isomorphism 085PND4KNT
• Ring 0894J980RK
• Class equation 08BPSC6YTE
• Product of primitive polynomials is primitive 08C1NWBNHA
• Product of cyclic groups is cyclic iff the orders are coprime 08FDFEF8D1
• Homomorphisms preserve identity and inverses 08FKQD7NJ3
• Fixed points of an action 08JF1K4DW1
• Lower central series 08PA437800
• Product formula for the order of HK 08PY4GNAPK
• Finite field 08RTDHX1SK
• Commutator subgroup 08SJ5K2CBG
• Order of an element divides the order of the group 08V7JHS3QZ
• Content of a polynomial 08VV7YJRB7
• Subfields of a finite field 08YERZDM9X
• In a PID a nonzero element is irreducible iff it generates a maximal ideal 095ZHQ7WCJ
• Subnormal series 096HN851YZ
• Fixed points of a p-group action 098D2E7DEH
• Algebraic extension 09EYMKMSB9
• Second Sylow theorem 09GCQ5STVJ
• Dihedral group 09GK3DE0WV
• The symmetric group S3 09H9Y0Q11W
• Outer automorphism group 09HEXDQ0CQ
• Maximal ideal 09J6KGT65G
• Groups of order p squared are abelian 09JZ9P4899
• Ring homomorphisms from a field are injective 09N7DZ7SDB
• Integral domain 09S3PFBEPM
• The number of Sylow subgroups is the index of the normaliser 09ZJ6CAK7S
• Tower law 0A7KJ68XHT
• Kernel of a ring homomorphism 0AAHE7GEW6
• Formal derivative 0AD4VS02H3
• If G modulo its centre is cyclic then G is abelian 0AH9Q6YNMT
• The symmetric group is generated by transpositions 0AQNX7G443
• Simplicity of the alternating group 0ARPRJQWZ2
• Power sum symmetric polynomials 0AS1FZT6BM
• Normaliser 0AT1ZSNM25
• R/P is an integral domain iff P is prime 0AVXBEC05W
• K[x]/(f) is a field iff f is irreducible 0AWMPX35CW
• Action on left cosets 0AZG100DKH
• Faithful action 0B0WE1GKTK
• R[x] is an integral domain iff R is 0B5Z1EC66J
• Ring isomorphism 0B6QC0C0G1
• Cyclic group 0B85Q41RP6
• Lattice of subgroups 0B8BW4YWTF
• Cycle type 0BCHJ30REG
• Order of the Galois group of a Galois extension equals its degree 0BDJBH8B4Z
• Monic polynomial 0BDX4MBKT4
• Additive group of integers 0BSCJA5G53
• Z[√-5] is not a UFD 0BSJYCG7NM
• Order of the dihedral group 0BZVNWJ352
• Simple group 0C093D1T4M
• The kernel is a normal subgroup 0C9ZKYSPYR
• Normal extension 0CBRFQ7DVZ
• Discriminant of a polynomial 0CCPN09EMJ
• A finite field of order q is the splitting field of x^q - x 0CDAHW576M
• Universal property of the free group 0CDWKDYWN6
• Polynomials in several variables 0CES91M79A
• Irreducible elements need not be prime 0CHE3J7QMX
• Finite subgroups of the multiplicative group of a field are cyclic 0CKGBEJX6B
• Impossibility of doubling the cube 0CQ357M0QS
• Automorphism group 0CWXCWP0Y0
• First isomorphism theorem 0CXXJNG5B6
• Correspondence theorem for rings 0D1R4SVMBQ
• Conjugate elements 0D1VWQGZZ7
• Polynomial ring over a UFD is a UFD 0D6KH8Y59N
• Conjugate subgroup 0D6WHTTH7T
• Uniqueness of inverses 0D9NDGE9KP
• Two-sided ideal 0DDA6CY7GJ
• Finite extensions are algebraic 0DFYTNZ6AM
• Order of a permutation is the lcm of its cycle lengths 0DH7HMXP9F
• Upper central series 0DJ9Y9V610
• Cosets have the same cardinality as the subgroup 0DKMBWTD7P
• Degree of a polynomial 0DPF6WRZW6
• K[x] is a Euclidean domain 0DQG39K2NH
• Universal property of the polynomial ring 0DQMDSVSXJ
• Galois extension 0DS2F7GK05
• Nilpotent group 0DZPD2MG33
• Jordan–Hölder theorem 0E088ZP533
• Orbit-stabiliser theorem 0E2HSCKZ6C
• A finite integral domain is a field 0EBE031B8E
• The multiplicative group of a finite field is cyclic 0EBZ3G455G
• Universal property of the abelianisation 0EJ4PC8NJ6
• Existence of an algebraic closure 0ETGBAK2CC
• Impossibility of trisecting the angle 0EWXPZCSZ0
• Prime elements are irreducible in an integral domain 0F0K8DB9TD
• g to the n equals 1 iff the order divides n 0F5QC101Q7
• Greatest common divisor in an integral domain 0F7KNJ1ZYN
• Exponent of a group 0FCCTFA275
• A subgroup whose index is the smallest prime divisor of the order is normal 0FEBE9NC0E
• Burnside's p^a q^b theorem 0FMAQSB1SC
• Cyclic group of order n 0G0B3DRG65
• Transcendence of pi 0G0F7FGCGP
• Automorphism group of a cyclic group 0G4C8SJ6PV
• The quaternions 0GE4DVQKR8
• Ring homomorphism 0GGE0Z0JQ1
• Frobenius endomorphism 0GHDDVAH8P
• Euclidean domain 0GJ2HRG95T
• Polynomial evaluation 0GJ7T0NK49
• Universal property of the quotient group 0GJ83E78A5
• Cancellation in an integral domain 0GJM24EXP9
• Minimal polynomial of an algebraic element 0GK1WB9R98
• The alternating group A5 is simple 0GKQKF0FXJ
• Nilpotent groups are solvable 0GNG2KVPZJ
• Degree of a field extension 0GS74FGFEK
• Fundamental theorem of symmetric polynomials 0GW1ZZH0R4
• Commutative ring 0GYGHKX834
• Inner automorphism 0H47DMFN3S
• The order of a finite field is a prime power 0H5WWC8RZX
• Abelianisation 0H9K5D3RDV
• Second isomorphism theorem for rings 0HJ4DTW6FN
• Group homomorphism 0HV9Y7WG5R
• Finitely generated group 0HY15D6WTX
• Conjugacy classes of the symmetric group correspond to cycle types 0J438GAC8K
• The symmetric group on five or more letters is not solvable 0J53NE674D
• Sign is a homomorphism 0JEJJCB0EX
• The ring of integers 0JHM6JK7FD
• Abelian simple groups have prime order 0JJV80RXHD
• Cauchy's theorem 0JM7PA288B
• Subgroups of abelian groups are normal 0JPDTEDCFD
• Order of an element 0JPYSD5E0B
• Order of the general linear group over a finite field 0JQ1KNQW2K
• Sylow p-subgroup 0JWZX6FKSH
• Galois group of a polynomial 0JZY3R908H
• Zero divisor 0K26YKNBSA
• Factor theorem 0K3MWA7DMC
• Action by left multiplication 0K5TWJAGPD
• Gaussian integers 0K5W69AC0J
• Subgroups of cyclic groups are cyclic 0KB1ND7XDS
• The alternating group is generated by three-cycles 0KFZ0QX83G
• A polynomial over a field has at most as many roots as its degree 0KJVQV3E0F
• Fixed field 0KMJ6919WS
• Reduction modulo p test for irreducibility 0KWWEXW0RN
• The number of Sylow subgroups divides the index 0M3PYJ3TSE
• The only ideals of a field are zero and the whole field 0M6G75E95E
• Adjoining a root of a polynomial 0M893X7YY7
• Vieta's formulas 0M8EWQJ1YV
• Field generated by adjoining elements 0MA5B744K3
• Sum of ideals 0MBJVDSK1R
• Generators of a group 0MBV2VHMXA
• A finite abelian group is cyclic iff its exponent equals its order 0MC0AQZSHE
• Algebraic closure 0MCXX29HWY
• Coprime ideals 0MDJP82TEH
• Division algorithm for polynomials 0MES3KS0ZC
• A ring homomorphism is injective iff its kernel is zero 0MMCB8RXGA
• Prime element 0MNHXBSX3M
• Product of rings 0MNZCN4BQK
• A quotient is abelian iff the subgroup contains the commutator subgroup 0MS8QGA6S5
• Noetherian ring 0MVH4Y1CRZ
• Ideal 0MYSA6MYNA
• Separable extension 0N019F6RE9
• Subgroup 0N0PXCTYA7
• A homomorphism is injective iff its kernel is trivial 0N2P0XRTDF
• Degree of a simple extension is the degree of the minimal polynomial 0N5QW9K7MH
• Nilpotent element 0N9R5NN6TW
• Composition series 0NB8VPB93V
• The minimal polynomial is irreducible 0NFEG48RDR
• Algebraic element 0NPW1DWECM
• Quaternion group 0NRJGW864D
• Finite extension 0NRYPM9Y27
• Product of two subgroups 0NS1YTGM19
• Z/pZ is a field for p prime 0NV6E43P4A
• Leading coefficient 0P05RJC5XA
• Solvable group 0P3KZK0FTV
• Simple extension 0P6SBR7YM2
• Characteristic of a ring 0P7CH1CND5
• Action by conjugation 0P8CY9V437
• Order of the symmetric group 0PCKBBMEXC
• Converse of Lagrange's theorem fails for the alternating group A4 0PDBVPP233
• Klein four-group 0PMRQJVTME
• Order of an inverse 0PMT9YJZVW
• The Gaussian integers are a Euclidean domain 0PN13BB7BT
• The constructible numbers form a field closed under square roots 0PN9WMRWT7
• Second isomorphism theorem 0PPP2J2DJV
• Centraliser 0Q0Z1BEXTF
• Internal direct product 0Q2MVQ7JMT
• Conjugacy class 0Q88KQD17A
• Third Sylow theorem 0Q8DGAZSJ4
• Polynomial ring 0Q9EZS90GY
• Field 0QA8CHH8MG
• The algebraic numbers form a field 0QFTETCHKB
• Gauss's lemma 0QFV5Y1Z5X
• Euclidean domains are principal ideal domains 0QKVE1GD9A
• The Galois group has order at most the degree 0QM98DNWVW
• Transitive action 0QP2YYR79N
• Von Dyck's theorem 0QPYHHNRXC
• In a UFD irreducible elements are prime 0QRCFZKXH8
• The group of units of Z/pZ is cyclic 0QTHB6PE52
• Frattini argument 0QTZDKJEDV
• Every element of a finite field of order q satisfies x^q = x 0QWR8WTZEW
• Presentation of a group 0QZCY9PE3Q
• Abelian groups are solvable 0R0X73XHAT
• Derived series 0R3H35G96G
• Universal property of the quotient ring 0R74MZYNRB
• Composition factors 0RFP7F3SEW
• Subgroup generated by a subset 0RG9SPQSWM
• Stabiliser 0RGGF8ZS6D
• Torsion subgroup 0RH1TPDGCQ
• Symmetric group 0RJMNEN3BJ
• Wedderburn's little theorem 0RRNVX1VKA
• Structure theorem for finitely generated abelian groups 0RY0B72XAS
• A nontrivial finite p-group has nontrivial centre 0RZCTT30H5
• Principal ideal domain 0RZYDVZ475
• Size of an orbit times the size of the stabiliser 0S19CWFZM2
• Order of the group of units of Z/nZ is Euler's totient 0S7495H4HM
• Centre of a group 0S7Y9Q990Q
• Correspondence theorem 0S9ZVZ34MD
• Order of a group 0SEX9F9J5Z
• Split polynomial 0SHMC00RJV
• Irreducibility of the cyclotomic polynomials over the rationals 0SK1VHZ5R2
• Sign of a permutation 0SK90238BH
• Primitive polynomial 0SP14DZN0G
• Subgroups of the integers 0SRATCFF1W
• Characteristic subgroups are normal 0SRE7K4E49
• The degree of a constructible number is a power of two 0SSSM1D475
• Primitive element theorem 0SVX98HYV1
• Order of the alternating group 0SZ1KMNWVE
• Normal core 0T0KMZPYV9
• Alternating group 0T13HM377F
• Fundamental theorem of Galois theory 0T1Q8D0FW8
• Degree of a product 0T233AZ1GE
• Existence of finite fields of every prime power order 0T8M3YC5RG
• Remainder theorem 0TEDQH56RW
• Third isomorphism theorem 0TJ1V9DDPC
• An algebraic extension of an algebraic extension is algebraic 0TPWS0K1BC
• Product of a subgroup and a normal subgroup is a subgroup 0TXWV1JGG1
• Sign of a cycle 0V0E04DWX2
• Subgroups of index two are normal 0V547YQRMV
• First Sylow theorem 0V98WY94B8
• Field of fractions 0VF11ZTTMD
• Quotient ring 0VFW4Y69BT
• Every polynomial has a root in some extension field 0VG02M1FEZ
• Size of a conjugacy class is the index of the centraliser 0VK5V9P8G3
• Polynomials of degree two or three are irreducible iff they have no roots 0VNE161QCK
• Frobenius automorphism of a finite field 0VP4TBVJ19
• Prime subfield 0VPFBAY7MW
• Cancellation laws 0VSNX4P2Q0
• Order of a power 0VVPQNVM8S
• Bézout's identity in a Euclidean domain 0VY3WPP9Y3
• Cyclotomic polynomial 0W0KVBM54S
• Normal subgroup 0W77CVBCHQ
• Chinese remainder theorem for rings 0W97J8QED7
• R/M is a field iff M is maximal 0WD3SE7S1X
• Newton's identities 0WG2A3HE0S
• Field extension 0WQ1A8SHTZ
• Group of nth roots of unity 0WX11EA3F3
• Division ring 0WZXK05ME7
• Idempotent element 0X4JKZXJ86
• Impossibility of squaring the circle 0X7CVDE3BG
• Number of elements of each order in a cyclic group 0XCYGCCX8V
• Decomposition of a permutation into transpositions 0XK6FECMW5
• Galois group 0XVR72M51M
• Gauss–Wantzel theorem 0Y34JMJNE9
• Primitive element of a finite field 0YAA9CJQXF
• Groups of order pq 0YBR6QPY4Y
• Cayley graph 0YE597NHGB
• Integers modulo n as a quotient group 0YJ3WRR9RY
• Hilbert's basis theorem 0YKA9J3XEE
• Free abelian group 0YMSEZE1VH
• Direct product of groups 0YP7WYXF9V
• Group action 0YSE802YCQ
• Cyclic subgroup generated by an element 0YT586TKXF
• Decomposition of a permutation into disjoint cycles 0YWDAZPRPQ
• Quotient group 0YYWCNTPWR
• Determinant as a group homomorphism 0YZ3T5Q7XN
• Kernel of a homomorphism 0Z424HK8CQ
• Burnside's lemma 0Z47M1B2VS
• Orbits partition the set 0ZC7FYE0NZ
• g to the order of the group is the identity 0ZEPNS9BXK
• Elementary symmetric polynomials 0ZMY8441GW
• First isomorphism theorem for rings 0ZMYX7GMH0
• K(a) is isomorphic to K[x] modulo the minimal polynomial 0ZN39QDTP9
• Tower law for indices 0ZQJ7KEFQ5
• Intersection of subgroups is a subgroup 0ZTGMDNVTB
• Lagrange's theorem 3C53TD