Math 2.0

Subjects

12 Field theory and polynomials

See MSC 12 on zbMATH Open.

NameTagKindMSC
Primitive element of a finite field00QSK85Y2Xconcept12E20
Transcendental element00SW8B91E7definition12F20
Irreducible polynomial00X2A9G9W3definition12E05
Ordered field010JCV9J4Edefinition12J15
Uniqueness of finite fields of a given order010PKCQPXPstatement12E20
Existence of splitting fields01441KDGBYstatement12F05
The complex numbers cannot be ordered01CJRJKDAAstatement12J15
Uniqueness of the algebraic closure01VQ7C4FTBstatement12F05
Eisenstein's criterion02BPKD7TMDstatement12E05
Intermediate field02NBKVQFRQdefinition12F05
Splitting field02NZZ1J0FHdefinition12F05
Algebraically closed field02SJG1HC32definition12F05
Irreducible polynomial03FKX7Z5DCconcept12E05
Separable polynomial03J8BS7FMVdefinition12F10
Field04C5ZBG1WTconcept12E99
Galois field04SGPNT8R9object12E20
Field generated by adjoining elements05NAS7AGMZconcept12F05
Uniqueness of splitting fields069A3BC3EPstatement12F05
The minimal polynomial divides every polynomial vanishing at the element06K7SARZKRstatement12F05
i squared is minus one07KBQCG9KVstatement12D99
Subfield07QY5RPFBRdefinition12F99
Rational root theorem0819CQ1V8Hstatement12E05
Finite field08RTDHX1SKdefinition12E20
Normal extension08S0NW8SESconcept12F10
Subfields of a finite field08YERZDM9Xstatement12E20
Algebraic extension09EYMKMSB9definition12F05
Ring homomorphisms from a field are injective09N7DZ7SDBstatement12E99
The rationals form an ordered field09RFKMBQ63statement12J15
Archimedean property0A74SX55QPstatement12J15
Tower law0A7KJ68XHTstatement12F05
Complete ordered field0A9E6X0S3Econcept12J15
K[x]/(f) is a field iff f is irreducible0AWMPX35CWstatement12E05
Order of the Galois group of a Galois extension equals its degree0BDJBH8B4Zstatement12F10
Finite extension0C1Y04SZ1Wconcept12F05
Normal extension0CBRFQ7DVZdefinition12F10
Discriminant of a polynomial0CCPN09EMJdefinition12E05
A finite field of order q is the splitting field of x^q - x0CDAHW576Mstatement12E20
Frobenius automorphism of a finite field0CGCPX8Z70concept12E20
Finite subgroups of the multiplicative group of a field are cyclic0CKGBEJX6Bstatement12E20
Impossibility of doubling the cube0CQ357M0QSstatement12F05
Finite extensions are algebraic0DFYTNZ6AMstatement12F05
Galois extension0DS2F7GK05definition12F10
Division ring0EA5BTA0CXconcept12E15
Degree of a field extension0EAJ2XXADCconcept12F05
The multiplicative group of a finite field is cyclic0EBZ3G455Gstatement12E20
Field extension0EQWVHF3W5concept12F05
Imaginary unit0ESCX36MMCobject12D99
Existence of an algebraic closure0ETGBAK2CCstatement12F05
Impossibility of trisecting the angle0EWXPZCSZ0statement12F05
The reals form a field0FGENPMPJXstatement12D99
Constructible number0FMN2MN7TMdefinition12F05
Inverse of a complex number0FQRAV5SKHstatement12D99
The complex numbers form a field0FRSR10DWFstatement12D99
Uniqueness of the complete ordered field0G8G1TH5WKstatement12J15
Subfield0GFGHRT2FJconcept12F99
Minimal polynomial of an algebraic element0GK1WB9R98definition12F05
Degree of a field extension0GS74FGFEKdefinition12F05
The order of a finite field is a prime power0H5WWC8RZXstatement12E20
Algebraic element0H628RSGWXconcept12F05
Archimedean ordered field0HQARMKEV5definition12J15
Fixed field0J1E5ZRZRMconcept12F10
Fundamental theorem of algebra0JP0W07Y9Astatement12D10
Split polynomial0JVSNJCN68concept12F05
Galois group of a polynomial0JZY3R908Hdefinition12F10
A polynomial over a field has at most as many roots as its degree0KJVQV3E0Fstatement12E05
Fixed field0KMJ6919WSdefinition12F10
Reduction modulo p test for irreducibility0KWWEXW0RNstatement12E05
Splitting field0KXQKX9P63concept12F05
Quadratic formula0KYZXYDGQZstatement12D10
The only ideals of a field are zero and the whole field0M6G75E95Estatement12E99
Adjoining a root of a polynomial0M893X7YY7definition12F05
Vieta's formulas0M8EWQJ1YVstatement12D10
Field generated by adjoining elements0MA5B744K3definition12F05
Algebraic closure0MCXX29HWYdefinition12F05
Separable extension0N019F6RE9definition12F10
The rationals form a field0N2WPCC00Tstatement12E99
Degree of a simple extension is the degree of the minimal polynomial0N5QW9K7MHstatement12F05
Simple extension0N681SSTHCconcept12F05
The minimal polynomial is irreducible0NFEG48RDRstatement12F05
Algebraic element0NPW1DWECMdefinition12F05
Transcendental element0NQMZ89X4Aconcept12F20
Finite extension0NRYPM9Y27definition12F05
Adjoining a root of a polynomial0NXWQK2Y56concept12F05
Ordered field0P3VCBDEE2concept12J15
Simple extension0P6SBR7YM2definition12F05
Algebraically closed field0PF6TM8TWTconcept12F05
The constructible numbers form a field closed under square roots0PN9WMRWT7statement12F05
Algebraic extension0PTBM7K0R2concept12F05
Galois group of a polynomial0Q5NA8Y362concept12F10
Field0QA8CHH8MGdefinition12E99
The Galois group has order at most the degree0QM98DNWVWstatement12F10
Galois extension0QMZAQ6QQ4concept12F10
Constructible number0QNFRBJF4Qconcept12F05
The reals form an ordered field0QSPNASE50statement12J15
Every element of a finite field of order q satisfies x^q = x0QWR8WTZEWstatement12E20
Wedderburn's little theorem0RRNVX1VKAstatement12E15
Separable polynomial0RZ89QWTSJconcept12F10
Split polynomial0SHMC00RJVdefinition12F05
Differentiation as a linear map0SS770NS8Bexample15A04, 12E10
The degree of a constructible number is a power of two0SSSM1D475statement12F05
Primitive element theorem0SVX98HYV1statement12F10
Fundamental theorem of Galois theory0T1Q8D0FW8statement12F10
Existence of finite fields of every prime power order0T8M3YC5RGstatement12E20
Prime subfield0TNS65QB87concept12E99
An algebraic extension of an algebraic extension is algebraic0TPWS0K1BCstatement12F05
Every polynomial has a root in some extension field0VG02M1FEZstatement12F05
Polynomials of degree two or three are irreducible iff they have no roots0VNE161QCKstatement12E05
Frobenius automorphism of a finite field0VP4TBVJ19definition12E20
Prime subfield0VPFBAY7MWdefinition12E99
Discriminant of a polynomial0VVK64Q4JKconcept12E05
Minimal polynomial (field theory)0VX9EP2GS8concept12F05
Separable extension0WCX7NB4N7concept12F10
Field extension0WQ1A8SHTZdefinition12F05
Division ring0WZXK05ME7definition12E15
Algebraic closure0X6AB5P3G1concept12F05
Impossibility of squaring the circle0X7CVDE3BGstatement12F05
Archimedean ordered field0XH79S06J3concept12J15
Galois group0XTY86BZKYconcept12F10
Galois group0XVR72M51Mdefinition12F10
Gauss–Wantzel theorem0Y34JMJNE9statement12F05
Primitive element of a finite field0YAA9CJQXFdefinition12E20
Existence of nth roots of complex numbers0YM6P6C206statement12D99
Intermediate field0Z409N8RDGconcept12F05
K(a) is isomorphic to K[x] modulo the minimal polynomial0ZN39QDTP9statement12F05
Finite field0ZXZ82FTX2concept12E20