Splitting field, the Glossary
In abstract algebra, a splitting field of a polynomial with coefficients in a field is the smallest field extension of that field over which the polynomial splits, i.e., decomposes into linear factors.[1]
Table of Contents
54 relations: Abstract algebra, Algebraic closure, Algebraically closed field, Augustin-Louis Cauchy, Basis (linear algebra), Bijection, Characteristic (algebra), Circular definition, Coefficient, Complex number, Comptes rendus de l'Académie des Sciences, Congruence relation, Converse (logic), Cube root, Degree of a field extension, Degree of a polynomial, Equivalence class, Field (mathematics), Field extension, Finite field, Galois extension, Galois group, Homomorphism, Ideal (ring theory), Indeterminate (variable), Injective function, Irreducible polynomial, Isomorphism, Mathematical proof, Maximal ideal, Minimal polynomial (field theory), Modular arithmetic, Monic polynomial, Parity (mathematics), Polynomial, Polynomial long division, Polynomial ring, Prime number, Quotient ring, Rational number, Real number, Residue field, Ring (mathematics), Ring homomorphism, Root of unity, Separable extension, Separable polynomial, Square (algebra), Surjective function, Total order, ... Expand index (4 more) »
Abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are sets with specific operations acting on their elements.
See Splitting field and Abstract algebra
Algebraic closure
In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed.
See Splitting field and Algebraic closure
Algebraically closed field
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in) has a root in. Splitting field and algebraically closed field are field (mathematics).
See Splitting field and Algebraically closed field
Augustin-Louis Cauchy
Baron Augustin-Louis Cauchy (France:, ; 21 August 1789 – 23 May 1857) was a French mathematician, engineer, and physicist.
See Splitting field and Augustin-Louis Cauchy
Basis (linear algebra)
In mathematics, a set of vectors in a vector space is called a basis (bases) if every element of may be written in a unique way as a finite linear combination of elements of.
See Splitting field and Basis (linear algebra)
Bijection
A bijection, bijective function, or one-to-one correspondence between two mathematical sets is a function such that each element of the first set (the domain) is mapped to exactly one element of the second set (the codomain).
See Splitting field and Bijection
Characteristic (algebra)
In mathematics, the characteristic of a ring, often denoted, is defined to be the smallest positive number of copies of the ring's multiplicative identity that will sum to the additive identity. Splitting field and characteristic (algebra) are field (mathematics).
See Splitting field and Characteristic (algebra)
Circular definition
A circular definition is a type of definition that uses the term(s) being defined as part of the description or assumes that the term(s) being described are already known.
See Splitting field and Circular definition
Coefficient
In mathematics, a coefficient is a multiplicative factor involved in some term of a polynomial, a series, or an expression.
See Splitting field and Coefficient
Complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted, called the imaginary unit and satisfying the equation i^.
See Splitting field and Complex number
Comptes rendus de l'Académie des Sciences
(English: Proceedings of the Academy of Sciences), or simply Comptes rendus, is a French scientific journal published since 1835.
See Splitting field and Comptes rendus de l'Académie des Sciences
Congruence relation
In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure in the sense that algebraic operations done with equivalent elements will yield equivalent elements.
See Splitting field and Congruence relation
Converse (logic)
In logic and mathematics, the converse of a categorical or implicational statement is the result of reversing its two constituent statements.
See Splitting field and Converse (logic)
Cube root
In mathematics, a cube root of a number is a number such that.
See Splitting field and Cube root
Degree of a field extension
In mathematics, more specifically field theory, the degree of a field extension is a rough measure of the "size" of the field extension.
See Splitting field and Degree of a field extension
Degree of a polynomial
In mathematics, the degree of a polynomial is the highest of the degrees of the polynomial's monomials (individual terms) with non-zero coefficients.
See Splitting field and Degree of a polynomial
Equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes.
See Splitting field and Equivalence class
Field (mathematics)
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers.
See Splitting field and Field (mathematics)
Field extension
In mathematics, particularly in algebra, a field extension (denoted L/K) is a pair of fields K \subseteq L, such that the operations of K are those of L restricted to K. In this case, L is an extension field of K and K is a subfield of L. For example, under the usual notions of addition and multiplication, the complex numbers are an extension field of the real numbers; the real numbers are a subfield of the complex numbers.
See Splitting field and Field extension
Finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements.
See Splitting field and Finite field
Galois extension
In mathematics, a Galois extension is an algebraic field extension E/F that is normal and separable; or equivalently, E/F is algebraic, and the field fixed by the automorphism group Aut(E/F) is precisely the base field F. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory.
See Splitting field and Galois extension
Galois group
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension.
See Splitting field and Galois group
Homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces).
See Splitting field and Homomorphism
Ideal (ring theory)
In mathematics, and more specifically in ring theory, an ideal of a ring is a special subset of its elements.
See Splitting field and Ideal (ring theory)
Indeterminate (variable)
In mathematics, particularly in formal algebra, an indeterminate is a symbol that is treated as a variable, but does not stand for anything else except itself.
See Splitting field and Indeterminate (variable)
Injective function
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies.
See Splitting field and Injective function
Irreducible polynomial
In mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials.
See Splitting field and Irreducible polynomial
Isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping.
See Splitting field and Isomorphism
Mathematical proof
A mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion.
See Splitting field and Mathematical proof
Maximal ideal
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all proper ideals.
See Splitting field and Maximal ideal
Minimal polynomial (field theory)
In field theory, a branch of mathematics, the minimal polynomial of an element of an extension field of a field is, roughly speaking, the polynomial of lowest degree having coefficients in the smaller field, such that is a root of the polynomial. Splitting field and minimal polynomial (field theory) are field (mathematics).
See Splitting field and Minimal polynomial (field theory)
Modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus.
See Splitting field and Modular arithmetic
Monic polynomial
In algebra, a monic polynomial is a non-zero univariate polynomial (that is, a polynomial in a single variable) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1.
See Splitting field and Monic polynomial
Parity (mathematics)
In mathematics, parity is the property of an integer of whether it is even or odd.
See Splitting field and Parity (mathematics)
Polynomial
In mathematics, a polynomial is a mathematical expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication and exponentiation to nonnegative integer powers, and has a finite number of terms.
See Splitting field and Polynomial
Polynomial long division
In algebra, polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree, a generalized version of the familiar arithmetic technique called long division.
See Splitting field and Polynomial long division
Polynomial ring
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, often a field.
See Splitting field and Polynomial ring
Prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers.
See Splitting field and Prime number
Quotient ring
In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra.
See Splitting field and Quotient ring
Rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator. Splitting field and rational number are field (mathematics).
See Splitting field and Rational number
Real number
In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a distance, duration or temperature.
See Splitting field and Real number
Residue field
In mathematics, the residue field is a basic construction in commutative algebra.
See Splitting field and Residue field
Ring (mathematics)
In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist.
See Splitting field and Ring (mathematics)
Ring homomorphism
In mathematics, a ring homomorphism is a structure-preserving function between two rings.
See Splitting field and Ring homomorphism
Root of unity
In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power.
See Splitting field and Root of unity
Separable extension
In field theory, a branch of algebra, an algebraic field extension E/F is called a separable extension if for every \alpha\in E, the minimal polynomial of \alpha over is a separable polynomial (i.e., its formal derivative is not the zero polynomial, or equivalently it has no repeated roots in any extension field).
See Splitting field and Separable extension
Separable polynomial
In mathematics, a polynomial P(X) over a given field K is separable if its roots are distinct in an algebraic closure of K, that is, the number of distinct roots is equal to the degree of the polynomial. Splitting field and separable polynomial are field (mathematics).
See Splitting field and Separable polynomial
Square (algebra)
In mathematics, a square is the result of multiplying a number by itself.
See Splitting field and Square (algebra)
Surjective function
In mathematics, a surjective function (also known as surjection, or onto function) is a function such that, for every element of the function's codomain, there exists one element in the function's domain such that.
See Splitting field and Surjective function
Total order
In mathematics, a total order or linear order is a partial order in which any two elements are comparable.
See Splitting field and Total order
Up to
Two mathematical objects and are called "equal up to an equivalence relation ".
Vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', can be added together and multiplied ("scaled") by numbers called ''scalars''.
See Splitting field and Vector space
Without loss of generality
Without loss of generality (often abbreviated to WOLOG, WLOG or w.l.o.g.; less commonly stated as without any loss of generality or with no loss of generality) is a frequently used expression in mathematics.
See Splitting field and Without loss of generality
Zero of a function
In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function f, is a member x of the domain of f such that f(x) vanishes at x; that is, the function f attains the value of 0 at x, or equivalently, x is a solution to the equation f(x).
See Splitting field and Zero of a function
References
[1] https://en.wikipedia.org/wiki/Splitting_field
Also known as Algebraic splitting field, Construction of splitting fields, Galois closure, Splitting feild, Splitting field of a polynomial.
, Up to, Vector space, Without loss of generality, Zero of a function.