English Dictionary

splitting field

noun

(algebra, Galois theory) (of a polynomial) Given a polynomial p over a field K, the smallest extension field L of K such that p, as a polynomial over L, decomposes into linear factors (polynomials of degree 1); (of a set of polynomials) given a set P of polynomials over K, the smallest extension field of K over which every polynomial in P decomposes into linear factors.

The splitting field of the polynomial is a field extension of the base field.

意味（1）

(algebra, Galois theory) (of a polynomial) Given a polynomial p over a field K, the smallest extension field L of K such that p, as a polynomial over L, decomposes into linear factors (polynomials of degree 1); (of a set of polynomials) given a set P of polynomials over K, the smallest extension field of K over which every polynomial in P decomposes into linear factors.

Theorem 3.2. If K is a field and f∈K[x] has degree n>1, then there exists a splitting field F of f with [F:K]<n!

意味（2）

(algebra, ring theory, of a K-algebra) Given a finite-dimensional K-algebra (algebra over a field), an extension field whose every simple (indecomposable) module is absolutely simple (remains simple after the scalar field has been extended to said extension field).

The terminology splitting field of a K-algebra is motivated by the same terminology regarding a polynomial. A splitting field of a K-algebra A is a field extension K↦L such that A⊗_KL is split; in the special case A=K[x]/f(x) this is the same as a splitting field of the polynomial f(x).

意味（3）

(algebra, ring theory, of a central simple algebra) Given a central simple algebra A over a field K, another field, E, such that the tensor product A⊗E is isomorphic to a matrix ring over E.

Every finite dimensional central simple algebra has a splitting field: moreover, if said CSA is a division algebra, then a maximal subfield of it is a splitting field.

意味（4）

(algebra, character theory) (of a character χ of a representation of a group G) A field K over which a K-representation of G exists which includes the character χ; (of a group G) a field over which a K-representation of G exists which includes every irreducible character in G.

DEFINITION 2. A field K is called a splitting field of a character 𝜒 of a group G if 𝜒∈ operatorname Char_K(G), i.e., 𝜒 is afforded by a K-representation of G.

Let T be a representation of G affording the character 𝜒. It follows from Definition 2 that K is a splitting field of 𝜒 if and only if T is equivalent to 𝛥, where 𝛥 is a K-representation of G. In other words, K is a splitting field of a character 𝜒 if and only if a representation T affording 𝜒 is realized over K. Every character of G has a splitting field (for example, C is a splitting field of any character of G). If K is a splitting field of both characters 𝜒₁,𝜒₂, then K is a splitting field of 𝜒₁+𝜒₂, Therefore, in studying splitting fields, we may consider irreducible characters only.

DEFINITION 2. A field K is called a splitting field of a character 𝜒 of a group G if 𝜒∈ operatorname Char_K(G), i.e., 𝜒 is afforded by a K-representation of G. Let T be a representation of G affording the character 𝜒. It follows from Definition 2 that K is a splitting field of 𝜒 if and only if T is equivalent to 𝛥, where 𝛥 is a K-representation of G. In other words, K is a splitting field of a character 𝜒 if and only if a representation T affording 𝜒 is realized over K. Every character of G has a splitting field (for example, C is a splitting field of any character of G). If K is a splitting field of both characters 𝜒₁,𝜒₂, then K is a splitting field of 𝜒₁+𝜒₂, Therefore, in studying splitting fields, we may consider irreducible characters only. DEFINITION 3. A field K is called a splitting field of a group G if it is a splitting field for every 𝜒∈ operatorname Irr(G).