Integral domain

What depends on this

71 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
• 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
• Algebraically closed field 02SJG1HC32
• Norm on the Gaussian integers 03AN1V4ZFJ
• Galois field 04SGPNT8R9
• Irreducible element 052SDEPZH7
• 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
• Z[x] is not a principal ideal domain 06E3AG9TM2
• Units of the Gaussian integers 06RH4XAGRY
• Unique factorisation domain 06YMXJMXYS
• Product of primitive polynomials is primitive 08C1NWBNHA
• Content of a polynomial 08VV7YJRB7
• In a PID a nonzero element is irreducible iff it generates a maximal ideal 095ZHQ7WCJ
• 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
• A finite field of order q is the splitting field of x^q - x 0CDAHW576M
• 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
• 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
• Euclidean domain 0GJ2HRG95T
• Cancellation in an integral domain 0GJM24EXP9
• The order of a finite field is a prime power 0H5WWC8RZX
• Gaussian integers 0K5W69AC0J
• A polynomial over a field has at most as many roots as its degree 0KJVQV3E0F
• Reduction modulo p test for irreducibility 0KWWEXW0RN
• Adjoining a root of a polynomial 0M893X7YY7
• Algebraic closure 0MCXX29HWY
• Degree of a simple extension is the degree of the minimal polynomial 0N5QW9K7MH
• The minimal polynomial is irreducible 0NFEG48RDR
• Z/pZ is a field for p prime 0NV6E43P4A
• The Gaussian integers are a Euclidean domain 0PN13BB7BT
• Gauss's lemma 0QFV5Y1Z5X
• Euclidean domains are principal ideal domains 0QKVE1GD9A
• In a UFD irreducible elements are prime 0QRCFZKXH8
• 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
• Fundamental theorem of Galois theory 0T1Q8D0FW8
• Degree of a product 0T233AZ1GE
• Existence of finite fields of every prime power order 0T8M3YC5RG
• 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
• Bézout's identity in a Euclidean domain 0VY3WPP9Y3
• Impossibility of squaring the circle 0X7CVDE3BG
• Gauss–Wantzel theorem 0Y34JMJNE9
• Primitive element of a finite field 0YAA9CJQXF
• K(a) is isomorphic to K[x] modulo the minimal polynomial 0ZN39QDTP9