Commutative ring

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126 nodes use this node, directly or through other nodes.

• The integers are a Euclidean domain 000JFQGDXG
• The characteristic of an integral domain is zero or prime 00758QM6JJ
• Principal ideal 00A2EDFT9Q
• Transcendental element 00SW8B91E7
• Irreducible polynomial 00X2A9G9W3
• Uniqueness of finite fields of a given order 010PKCQPXP
• Existence of splitting fields 01441KDGBY
• Uniqueness of the algebraic closure 01VQ7C4FTB
• Euclidean algorithm in a Euclidean domain 023VY493K0
• Eisenstein's criterion 02BPKD7TMD
• Intermediate field 02NBKVQFRQ
• Splitting field 02NZZ1J0FH
• Algebraically closed field 02SJG1HC32
• Ideal generated by a subset 030TZ0DQWT
• Freshman's dream 0396VGFA78
• Norm on the Gaussian integers 03AN1V4ZFJ
• Divisibility in a commutative ring 04GNK7K4VP
• Galois field 04SGPNT8R9
• Irreducible element 052SDEPZH7
• Associate elements 053E86NWVK
• PID implies UFD 055HT85T3S
• The rationals as the field of fractions of the integers 05K3J8J8T2
• Maximal ideals are prime 05WAF5T0GZ
• Uniqueness of splitting fields 069A3BC3EP
• Symmetric polynomial 06CTFY2DEZ
• Z[x] is not a principal ideal domain 06E3AG9TM2
• Prime ideal 06HC6HK0Z8
• The minimal polynomial divides every polynomial vanishing at the element 06K7SARZKR
• Units of the Gaussian integers 06RH4XAGRY
• Unique factorisation domain 06YMXJMXYS
• Subfield 07QY5RPFBR
• Product of primitive polynomials is primitive 08C1NWBNHA
• Finite field 08RTDHX1SK
• 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
• Algebraic extension 09EYMKMSB9
• Ring homomorphisms from a field are injective 09N7DZ7SDB
• Integral domain 09S3PFBEPM
• Tower law 0A7KJ68XHT
• Power sum symmetric polynomials 0AS1FZT6BM
• R/P is an integral domain iff P is prime 0AVXBEC05W
• K[x]/(f) is a field iff f is irreducible 0AWMPX35CW
• R[x] is an integral domain iff R is 0B5Z1EC66J
• Order of the Galois group of a Galois extension equals its degree 0BDJBH8B4Z
• Z[√-5] is not a UFD 0BSJYCG7NM
• Normal extension 0CBRFQ7DVZ
• Discriminant of a polynomial 0CCPN09EMJ
• A finite field of order q is the splitting field of x^q - x 0CDAHW576M
• 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
• Polynomial ring over a UFD is a UFD 0D6KH8Y59N
• Finite extensions are algebraic 0DFYTNZ6AM
• K[x] is a Euclidean domain 0DQG39K2NH
• Galois extension 0DS2F7GK05
• A finite integral domain is a field 0EBE031B8E
• The multiplicative group of a finite field is cyclic 0EBZ3G455G
• Existence of an algebraic closure 0ETGBAK2CC
• Impossibility of trisecting the angle 0EWXPZCSZ0
• Prime elements are irreducible in an integral domain 0F0K8DB9TD
• Greatest common divisor in an integral domain 0F7KNJ1ZYN
• Transcendence of pi 0G0F7FGCGP
• Frobenius endomorphism 0GHDDVAH8P
• Euclidean domain 0GJ2HRG95T
• Cancellation in an integral domain 0GJM24EXP9
• Minimal polynomial of an algebraic element 0GK1WB9R98
• Degree of a field extension 0GS74FGFEK
• Fundamental theorem of symmetric polynomials 0GW1ZZH0R4
• The order of a finite field is a prime power 0H5WWC8RZX
• The ring of integers 0JHM6JK7FD
• Galois group of a polynomial 0JZY3R908H
• Gaussian integers 0K5W69AC0J
• 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 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
• Algebraic closure 0MCXX29HWY
• Prime element 0MNHXBSX3M
• Noetherian ring 0MVH4Y1CRZ
• Separable extension 0N019F6RE9
• Degree of a simple extension is the degree of the minimal polynomial 0N5QW9K7MH
• The minimal polynomial is irreducible 0NFEG48RDR
• Algebraic element 0NPW1DWECM
• Finite extension 0NRYPM9Y27
• Z/pZ is a field for p prime 0NV6E43P4A
• Simple extension 0P6SBR7YM2
• The Gaussian integers are a Euclidean domain 0PN13BB7BT
• The constructible numbers form a field closed under square roots 0PN9WMRWT7
• 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
• In a UFD irreducible elements are prime 0QRCFZKXH8
• Every element of a finite field of order q satisfies x^q = x 0QWR8WTZEW
• Wedderburn's little theorem 0RRNVX1VKA
• Principal ideal domain 0RZYDVZ475
• Irreducibility of the cyclotomic polynomials over the rationals 0SK1VHZ5R2
• Primitive polynomial 0SP14DZN0G
• The degree of a constructible number is a power of two 0SSSM1D475
• Primitive element theorem 0SVX98HYV1
• Fundamental theorem of Galois theory 0T1Q8D0FW8
• Degree of a product 0T233AZ1GE
• Existence of finite fields of every prime power order 0T8M3YC5RG
• An algebraic extension of an algebraic extension is algebraic 0TPWS0K1BC
• Field of fractions 0VF11ZTTMD
• Every polynomial has a root in some extension field 0VG02M1FEZ
• Polynomials of degree two or three are irreducible iff they have no roots 0VNE161QCK
• Frobenius automorphism of a finite field 0VP4TBVJ19
• Prime subfield 0VPFBAY7MW
• Bézout's identity in a Euclidean domain 0VY3WPP9Y3
• R/M is a field iff M is maximal 0WD3SE7S1X
• Newton's identities 0WG2A3HE0S
• Field extension 0WQ1A8SHTZ
• Impossibility of squaring the circle 0X7CVDE3BG
• Galois group 0XVR72M51M
• Gauss–Wantzel theorem 0Y34JMJNE9
• Primitive element of a finite field 0YAA9CJQXF
• Hilbert's basis theorem 0YKA9J3XEE
• Elementary symmetric polynomials 0ZMY8441GW
• K(a) is isomorphic to K[x] modulo the minimal polynomial 0ZN39QDTP9