Wiki143:WikiProject Mathematics/PlanetMath Exchange/12-XX Field theory and polynomials

12-00 General reference works (handbooks, dictionaries, bibliographies, etc.)

• PM: monomial, id=2824 -- WP guess: monomial -- Status:

• PM: order and degree of polynomial, id=4289 -- WP guess: order and degree of polynomial -- Status:

• PM: homogeneous polynomial, id=3872 -- WP guess: homogeneous polynomial -- Status:

• PM: weighted homogeneous polynomial, id=7177 -- WP guess: weighted homogeneous polynomial -- Status:

• PM: cardinality of monomials, id=8769^new! -- WP guess: cardinality of monomials -- Status:

12D05 Polynomials: factorization

• PM: factor theorem, id=1810 -- WP: factor theorem -- Status: NM

AdamSmithee. 18:22, 19 December 2005 (UTC)

• PM: polynomial long division, id=5803 -- WP guess: polynomial long division -- Status:

• PM: proof of factor theorem, id=2937 -- WP: factor theorem -- Status: NM

AdamSmithee. 18:22, 19 December 2005 (UTC)

• PM: proof of rational root theorem, id=3472 -- WP guess: proof of rational root theorem -- Status:

• PM: rational root theorem, id=228 -- WP guess: rational root theorem -- Status:

• PM: sextic equation, id=4682 -- WP guess: sextic equation -- Status: NC

AdamSmithee 21:31, 19 December 2005 (UTC)

• PM: synthetic division, id=5531 -- WP: synthetic division -- Status: A

AdamSmithee 21:31, 19 December 2005 (UTC)

• PM: proof of factor theorem due to Fermat, id=7610 -- WP guess: proof of factor theorem due to Fermat -- Status:

• PM: proof of factor theorem using division, id=6896 -- WP guess: proof of factor theorem using division -- Status:

12D10 Polynomials: location of zeros (algebraic theorems)

• PM: Cardano's derivation of the cubic formula, id=1408 -- WP: cubic equation -- Status: A

AdamSmithee 21:31, 19 December 2005 (UTC)

• PM: cubic formula, id=1407 -- WP: cubic equation -- Status: N

As the talk page suggests that the formula shouldn't be put in the article, I've put in an external link - AdamSmithee 21:58, 29 December 2005 (UTC)

• PM: derivation of quadratic formula, id=700 -- WP: quadratic equation -- Status: A

AdamSmithee 21:31, 19 December 2005 (UTC)

• PM: Ferrari-Cardano derivation of the quartic formula, id=2883 -- WP: quartic equation -- Status: A

AdamSmithee 21:31, 19 December 2005 (UTC)

• PM: Galois-theoretic derivation of the cubic formula, id=1438 -- WP: cubic equation -- Status: NM

AdamSmithee 21:31, 19 December 2005 (UTC)

• PM: Galois-theoretic derivation of the quartic formula, id=2825 -- WP: quartic equation -- Status: A

AdamSmithee 21:31, 19 December 2005 (UTC)

• PM: irreducible polynomial, id=5907 -- WP: irreducible polynomial -- Status: A

AdamSmithee 18:23, 19 December 2005 (UTC)

• PM: multiplicity, id=5905 -- WP: multiplicity -- Status: NM

AdamSmithee 21:31, 19 December 2005 (UTC)

• PM: properties of quadratic equation, id=6115 -- WP: quadratic equation -- Status: NM

AdamSmithee 21:31, 19 December 2005 (UTC)

• PM: quadratic formula, id=227 -- WP: quadratic equation -- Status: A

AdamSmithee 21:31, 19 December 2005 (UTC)

• PM: quartic formula, id=1525 -- WP: quartic equation -- Status: N

There is already an external link - AdamSmithee 21:58, 29 December 2005 (UTC)

• PM: reciprocal polynomial, id=4239 -- WP: reciprocal polynomial -- Status: NM

AdamSmithee 21:31, 19 December 2005 (UTC)

• PM: root, id=1307 -- WP: root (mathematics) -- Status: NM

AdamSmithee 21:31, 19 December 2005 (UTC)

• PM: variant of Cardano's derivation, id=4296 -- WP guess: variant of Cardano's derivation -- Status:

• PM: Cardano's formulae, id=7172 -- WP guess: Cardano's formulae -- Status:

• PM: irreducibility of binomials with unity coefficients, id=6982 -- WP guess: irreducibility of binomials with unity coefficients -- Status:

• PM: polynomial equation of odd degree, id=7588 -- WP guess: polynomial equation of odd degree -- Status:

• PM: trigonometric cubic formula, id=6746 -- WP guess: trigonometric cubic formula -- Status:

12D15 Fields related with sums of squares (formally real fields, Pythagorean fields, etc.)

• PM: Euclidean field, id=5869 -- WP guess: Euclidean field -- Status:

• PM: formally real field, id=5863 -- WP guess: formally real field -- Status:

• PM: positive cone, id=6430 -- WP guess: positive cone -- Status:

• PM: Pythagorean field, id=5868 -- WP guess: Pythagorean field -- Status:

• PM: stufe of a field, id=6829 -- WP guess: stufe of a field -- Status:

• PM: theorems on sums of squares, id=6830 -- WP guess: theorems on sums of squares -- Status:

• PM: constructible numbers, id=9583^new! -- WP guess: constructible numbers -- Status:

• PM: criterion for constructibility of regular polygon, id=9660^new! -- WP guess: criterion for constructibility of regular polygon -- Status:

• PM: motivation of definition of constructible numbers, id=9607^new! -- WP guess: motivation of definition of constructible numbers -- Status:

• PM: theorem on constructible angles, id=9605^new! -- WP guess: theorem on constructible angles -- Status:

• PM: theorem on constructible numbers, id=9614^new! -- WP guess: theorem on constructible numbers -- Status:

12D99 Miscellaneous

• PM: Archimedean property, id=3396 -- WP: Archimedean property -- Status: NM

The WP article is rather advanced. The PM article can be used to improve on WP - AdamSmithee 21:58, 29 December 2005 (UTC)

• PM: calculating the nth roots of a complex number, id=5667 -- WP guess: calculating the nth roots of a complex number -- Status:

• PM: complex, id=720 -- WP: complex number -- Status: A

AdamSmithee 21:58, 29 December 2005 (UTC)

• PM: complex conjugate, id=1508 -- WP: complex conjugate -- Status: A

AdamSmithee 21:58, 29 December 2005 (UTC)

• PM: complex number, id=471 -- WP: complex number -- Status: A

AdamSmithee 21:58, 29 December 2005 (UTC)

• PM: proof that every ordered field with the least upper bound property is isomorphic to \mathbb{R}, id=5607 -- WP guess: proof that every ordered field with the least upper bound property is isomorphic to \mathbbR -- Status:

• PM: every ordered field with the least upper bound property is isomorphic to the real numbers, id=5602 -- WP guess: every ordered field with the least upper bound property is isomorphic to the real numbers -- Status:

• PM: examples of totally real fields, id=4675 -- WP guess: examples of totally real fields -- Status:

• PM: extended real numbers, id=4441 -- WP guess: extended real numbers -- Status:

• PM: fundamental theorem of algebra, id=1927 -- WP: fundamental theorem of algebra -- Status: A

AdamSmithee 21:58, 29 December 2005 (UTC)

• PM: fundamental theorem of algebra result, id=5851 -- WP: fundamental theorem of algebra -- Status: A

AdamSmithee 21:58, 29 December 2005 (UTC)

• PM: imaginary, id=2017 -- WP guess: imaginary -- Status:

• PM: imaginary unit, id=2018 -- WP: imaginary unit -- Status: A

AdamSmithee 21:58, 29 December 2005 (UTC)

• PM: indeterminate form, id=2658 -- WP guess: indeterminate form -- Status:

• PM: inequalities for real numbers, id=4742 -- WP guess: inequalities for real numbers -- Status:

• PM: interval, id=4446 -- WP guess: interval -- Status:

• PM: Lindemann-Weierstrass theorem, id=5791 -- WP guess: Lindemann-Weierstrass theorem -- Status:

• PM: modulus of complex number, id=4242 -- WP: complex number -- Status: NM

Can be used for a better wording of section 1.3 - AdamSmithee 21:58, 29 December 2005 (UTC)

• PM: nth root, id=754 -- WP guess: nth root -- Status:

• PM: proof of fundamental theorem of algebra, id=3603 -- WP guess: proof of fundamental theorem of algebra -- Status:

• PM: proof of fundamental theorem of algebra (argument principle), id=6174 -- WP guess: proof of fundamental theorem of algebra (argument principle) -- Status:

• PM: proof of fundamental theorem of algebra (due to D'Alembert), id=6171 -- WP guess: proof of fundamental theorem of algebra (due to D'Alembert) -- Status:

• PM: proof of fundamental theorem of algebra (Rouché's theorem), id=6177 -- WP guess: proof of fundamental theorem of algebra (Rouché's theorem) -- Status:

• PM: proof of the fundamental theorem of algebra (Liouville's theorem), id=1928 -- WP guess: proof of the fundamental theorem of algebra (Liouville's theorem) -- Status:

• PM: proof of the fundamental theorem of algebra (using Brouwer's fixed point theorem), id=6187 -- WP guess: proof of the fundamental theorem of algebra (using Brouwer's fixed point theorem) -- Status:

• PM: proportion equation, id=6585 -- WP guess: proportion equation -- Status:

• PM: real and complex embeddings, id=4666 -- WP guess: real and complex embeddings -- Status:

• PM: real number, id=454 -- WP guess: real number -- Status:

• PM: totally real and imaginary fields, id=4673 -- WP guess: totally real and imaginary fields -- Status:

• PM: 0\ne 1 as real numbers, id=7218 -- WP guess: 0\ne 1 as real numbers -- Status:

• PM: axiomatic definition of the real numbers, id=7591 -- WP guess: axiomatic definition of the real numbers -- Status:

• PM: Hermite polynomials, id=7061 -- WP guess: Hermite polynomials -- Status:

• PM: index of inequalities, id=7629 -- WP guess: index of inequalities -- Status:

• PM: opposite number, id=6775 -- WP guess: opposite number -- Status:

• PM: quadratic inequality, id=7233 -- WP guess: quadratic inequality -- Status:

• PM: taking square root algebraically, id=7175 -- WP guess: taking square root algebraically -- Status:

• PM: topic entry on real numbers, id=7618 -- WP guess: topic entry on real numbers -- Status:

• PM: Archimedean ordered fields are real, id=9819^new! -- WP guess: Archimedean ordered fields are real -- Status:

• PM: continuous functions on the extended real numbers, id=9272^new! -- WP guess: continuous functions on the extended real numbers -- Status:

• PM: examples of continuous functions on the extended real numbers, id=9273^new! -- WP guess: examples of continuous functions on the extended real numbers -- Status:

• PM: existence of nth root, id=7867^new! -- WP guess: existence of nth root -- Status:

• PM: existence of nth root, id=8222^new! -- WP guess: existence of nth root -- Status:

• PM: infimum and supremum for real numbers, id=7638^new! -- WP guess: infimum and supremum for real numbers -- Status:

• PM: rational numbers are real numbers, id=7719^new! -- WP guess: rational numbers are real numbers -- Status:

• PM: solving certain polynomial inequalities, id=9226^new! -- WP guess: solving certain polynomial inequalities -- Status:

• PM: the groups of real numbers, id=8226^new! -- WP guess: the groups of real numbers -- Status:

12E05 Polynomials (irreducibility, etc.)

• PM: discriminant, id=2759 -- WP guess: discriminant -- Status:

• PM: Gauss's Lemma I, id=3566 -- WP guess: Gauss's Lemma I -- Status:

• PM: Gauss's Lemma II, id=3567 -- WP guess: Gauss's Lemma II -- Status:

• PM: resolvent, id=2792 -- WP guess: resolvent -- Status:

Planetmath link points on resultant. Wiki link is a disambiguation page.

Article on resolvent in Galois theory is lacking everywhere.

Resultant is lacking in this list. D.Lazard (talk) 11:49, 29 October 2011 (UTC)

• PM: Tchirnhaus transformations, id=4572 -- WP guess: Tchirnhaus transformations -- Status:

• PM: calculation of square root of polynomial, id=7430 -- WP guess: calculation of square root of polynomial -- Status:

• PM: partial fractions for polynomials, id=7613 -- WP guess: partial fractions for polynomials -- Status:

• PM: square root of polynomial, id=7428 -- WP guess: square root of polynomial -- Status:

• PM: examples of minimal polynomials, id=9184^new! -- WP guess: examples of minimal polynomials -- Status:

12E10 Special polynomials

• PM: de Moivre identity, id=1994 -- WP: De Moivre's formula -- Status: A

Oleg Alexandrov (talk) 18:56, 20 November 2005 (UTC)

• PM: monic, id=1899 -- WP guess: monic -- Status:

• PM: proof of de Moivre identity, id=6126 -- WP: De Moivre's formula -- Status:A

Oleg Alexandrov (talk) 18:56, 20 November 2005 (UTC)

• PM: all one polynomial, id=6712 -- WP guess: all one polynomial -- Status:

12E15 Skew fields, division rings

• PM: proof of Wedderburn's theorem, id=3627 -- WP: proof of Wedderburn's theorem -- Status: N

Oleg Alexandrov (talk) 19:05, 20 November 2005 (UTC)

• PM: second proof of Wedderburn's theorem, id=4198 -- WP guess: second proof of Wedderburn's theorem -- Status: N

Oleg Alexandrov (talk) 19:05, 20 November 2005 (UTC)

• PM: Wedderburn's Theorem, id=3473 -- WP guess: Division ring -- Status: A

Oleg Alexandrov (talk) 19:05, 20 November 2005 (UTC)

• PM: subfield criterion, id=8598^new! -- WP guess: subfield criterion -- Status:

12E20 Finite fields (field-theoretic aspects)

• PM: proof that Q is the prime subfield of any field of characteristic 0, id=7600 -- WP guess: proof that Q is the prime subfield of any field of characteristic 0 -- Status:

12E25 Hilbertian fields; Hilbert's irreducibility theorem

• PM: Hilbert's irreducibility theorem, id=7010 -- WP guess: Hilbert's irreducibility theorem -- Status:

• PM: thin algebraic set, id=7009 -- WP guess: thin algebraic set -- Status:

12E99 Miscellaneous

• PM: characteristic, id=1160 -- WP: characteristic (mathematics) -- Status:

• PM: characterization of field, id=4715 -- WP guess: characterization of field -- Status:

• PM: characterizations of integral, id=6642 -- WP guess: characterizations of integral -- Status:

• PM: example of an infinite field of finite characteristic, id=3635 -- WP guess: example of an infinite field of finite characteristic -- Status:

• PM: examples of fields, id=3162 -- WP: Field (mathematics) -- Status: A

AdamSmithee 14:08, 30 December 2005 (UTC)

• PM: field, id=355 -- WP: Field (mathematics) -- Status: A

AdamSmithee 14:08, 30 December 2005 (UTC)

• PM: field homomorphism, id=4670 -- WP: Ring homomorphism -- Status: NM

The WP article may use a separate section on fields; also, a redirect AdamSmithee 14:08, 30 December 2005 (UTC)

• PM: Frobenius homomorphism, id=2157 -- WP guess: Frobenius homomorphism -- Status:

• PM: groups in field, id=6311 -- WP guess: groups in field -- Status:

• PM: homomorphisms from fields are either injective or trivial, id=6243 -- WP: Ring homomorphism -- Status: NM

AdamSmithee 14:08, 30 December 2005 (UTC)

• PM: integral element, id=5715 -- WP guess: integral element -- Status:

• PM: integrity characterized by places, id=6643 -- WP guess: integrity characterized by places -- Status:

• PM: place as extension of homomorphism, id=6651 -- WP guess: place as extension of homomorphism -- Status:

• PM: place of field, id=6640 -- WP guess: place of field -- Status:

• PM: prime subfield, id=2892 -- WP: prime subfield -- Status: NM

AdamSmithee 14:08, 30 December 2005 (UTC)

• PM: subfield, id=5055 -- WP guess: subfield -- Status:

• PM: field homomorphisms fix prime subfields, id=8461^new! -- WP guess: field homomorphisms fix prime subfields -- Status:

12F05 Algebraic extensions

• PM: a finite extension of fields is an algebraic extension, id=4725 -- WP guess: a finite extension of fields is an algebraic extension -- Status:

• PM: algebraic closure, id=1510 -- WP guess: algebraic closure -- Status:

• PM: algebraic extension, id=4724 -- WP guess: algebraic extension -- Status:

• PM: algebraically closed, id=1509 -- WP guess: algebraically closed -- Status:

• PM: algebraically dependent, id=4741 -- WP guess: algebraically dependent -- Status:

• PM: existence of the minimal polynomial, id=4723 -- WP guess: existence of the minimal polynomial -- Status:

• PM: finite extension, id=708 -- WP guess: finite extension -- Status:

• PM: minimal polynomial, id=3850 -- WP guess: minimal polynomial -- Status:

• PM: norm, id=1846 -- WP guess: norm -- Status:

• PM: primitive element theorem, id=214 -- WP guess: primitive element theorem -- Status:

• PM: proof of primitive element theorem, id=3912 -- WP guess: proof of primitive element theorem -- Status:

• PM: proof of primitive element theorem, id=5725 -- WP guess: proof of primitive element theorem -- Status:

• PM: splitting field, id=1303 -- WP guess: splitting field -- Status:

• PM: Steinitz theorem, id=5985 -- WP guess: Steinitz theorem -- Status:

• PM: the field extension \mathbb{R}/\mathbb{Q} is not finite, id=4726 -- WP guess: the field extension \mathbbR/\mathbbQ is not finite -- Status:

• PM: trace, id=1845 -- WP guess: trace -- Status:

• PM: p-extension, id=6764 -- WP guess: p-extension -- Status:

• PM: Kummer theory, id=6793 -- WP guess: Kummer theory -- Status:

• PM: algebraic closure of a finite field, id=8889^new! -- WP guess: algebraic closure of a finite field -- Status:

• PM: alternative definition of algebraically closed, id=9145^new! -- WP guess: alternative definition of algebraically closed -- Status:

• PM: cardinality of algebraic closure, id=8649^new! -- WP guess: cardinality of algebraic closure -- Status:

• PM: every algebraically closed field is perfect, id=9139^new! -- WP guess: every algebraically closed field is perfect -- Status:

• PM: examples of trace and norm, id=7937^new! -- WP guess: examples of trace and norm -- Status:

• PM: expressible, id=9192^new! -- WP guess: expressible -- Status:

• PM: finite field cannot be algebraically closed, id=8654^new! -- WP guess: finite field cannot be algebraically closed -- Status:

• PM: integral closures in separable extensions are finitely generated, id=9323^new! -- WP guess: integral closures in separable extensions are finitely generated -- Status:

• PM: quadratic closure, id=7659^new! -- WP guess: quadratic closure -- Status:

• PM: quadratic extension, id=7656^new! -- WP guess: quadratic extension -- Status:

• PM: radical, id=9190^new! -- WP guess: radical -- Status:

• PM: relative algebraic closure, id=7939^new! -- WP guess: relative algebraic closure -- Status:

• PM: separability is required for integral closures to be finitely generated, id=9324^new! -- WP guess: separability is required for integral closures to be finitely generated -- Status:

• PM: separably algebraically closed field, id=7991^new! -- WP guess: separably algebraically closed field -- Status:

• PM: splitting field of a finite set of polynomials, id=9140^new! -- WP guess: splitting field of a finite set of polynomials -- Status:

12F10 Separable extensions, Galois theory

• PM: Abelian extension, id=3599 -- WP guess: Abelian extension -- Status:

• PM: absolute Galois group, id=1515 -- WP guess: absolute Galois group -- Status:

• PM: cyclic extension, id=3600 -- WP guess: cyclic extension -- Status:

• PM: example of nonperfect field, id=3580 -- WP guess: example of nonperfect field -- Status:

• PM: example of normal extension, id=6051 -- WP guess: example of normal extension -- Status:

• PM: fixed field, id=1325 -- WP guess: fixed field -- Status:

• PM: Galois closure, id=3601 -- WP guess: Galois closure -- Status:

• PM: Galois conjugate, id=1519 -- WP guess: Galois conjugate -- Status:

• PM: Galois extension, id=1326 -- WP guess: Galois extension -- Status:

• PM: Galois group, id=1317 -- WP guess: Galois group -- Status:

• PM: infinite Galois theory, id=2918 -- WP guess: infinite Galois theory -- Status:

• PM: normal closure, id=3602 -- WP guess: normal closure -- Status:

• PM: normal extension, id=1310 -- WP guess: normal extension -- Status:

• PM: perfect field, id=3577 -- WP guess: perfect field -- Status:

• PM: proof of characterization of perfect fields, id=6448 -- WP guess: proof of characterization of perfect fields -- Status:

• PM: radical extension, id=1329 -- WP guess: radical extension -- Status:

• PM: separable, id=1304 -- WP guess: separable -- Status:

• PM: separable closure, id=1514 -- WP guess: separable closure -- Status:

• PM: symmetric polynomial, id=1337 -- WP guess: symmetric polynomial -- Status:

• PM: casus irreducibilis, id=7171 -- WP guess: casus irreducibilis -- Status:

• PM: example of an extension that is not normal, id=8038^new! -- WP guess: example of an extension that is not normal -- Status:

• PM: Galois is not transitive, id=8041^new! -- WP guess: Galois is not transitive -- Status:

• PM: normal is not transitive, id=8043^new! -- WP guess: normal is not transitive -- Status:

• PM: solvable by radicals, id=9191^new! -- WP guess: solvable by radicals -- Status:

12F15 Inseparable extensions

• PM: purely inseparable, id=6480 -- WP guess: purely inseparable -- Status:

12F20 Transcendental extensions

• PM: linearly disjoint, id=5793 -- WP guess: linearly disjoint -- Status:

• PM: transcendence degree, id=4740 -- WP guess: transcendence degree -- Status:

12F99 Miscellaneous

• PM: composite field, id=1511 -- WP guess: composite field -- Status:

• PM: extension field, id=703 -- WP guess: extension field -- Status:

• PM: field adjunction, id=5875 -- WP guess: field adjunction -- Status:

• PM: p-adic canonical form, id=5665 -- WP guess: p-adic canonical form -- Status:

• PM: simple field extension, id=5878 -- WP guess: simple field extension -- Status:

• PM: non-constant element of rational function field, id=6762 -- WP guess: non-constant element of rational function field -- Status:

• PM: simple transcendental field extension, id=6751 -- WP guess: simple transcendental field extension -- Status:

• PM: lattice of fields, id=9550^new! -- WP guess: lattice of fields -- Status:

12H05 Differential algebra

• PM: differential field, id=5778 -- WP guess: differential field -- Status:

12J05 Normed fields

• PM: Gelfand-Tornheim theorem, id=5628 -- WP guess: Gelfand-Tornheim theorem -- Status:

12J10 Valued fields

• PM: complete ultrametric field, id=6615 -- WP guess: complete ultrametric field -- Status:

12J15 Ordered fields

• PM: some properties of uncountable subsets of the real numbers, id=8886^new! -- WP guess: some properties of uncountable subsets of the real numbers -- Status:

12J20 General valuation theory

• PM: discrete valuation, id=4761 -- WP guess: discrete valuation -- Status:

• PM: non-isomorphic completions of \mathbb{Q}, id=6671 -- WP guess: non-isomorphic completions of \mathbbQ -- Status:

• PM: value group of completion, id=6670 -- WP guess: value group of completion -- Status:

12J99 Miscellaneous

• PM: topological ring, id=3076 -- WP guess: topological ring -- Status:

12K10 Semifields

• PM: semifield, id=7718^new! -- WP guess: semifield -- Status:

12Y05 Computational aspects of field theory and polynomials

• PM: proof of Vieta's formula, id=7297 -- WP guess: proof of Vieta's formula -- Status:

• PM: Vieta's formula, id=7190 -- WP guess: Vieta's formula -- Status: