English-Finnish translations for splitting field

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 ; given a set ''P'of polynomials over ''K'', the smallest extension field of ''K'over which every polynomial in ''P'decomposes into linear factors
Given a finite-dimensional K-algebra|''K''-algebra , an extension field whose every simple module is absolutely simple (remains simple after the scalar field has been extended to said extension field
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''
A field ''K'over which a ''K''-representation of ''G'exists which includes the character χ; a field over which a ''K''-representation of ''G'exists which includes every irreducible character in ''G''

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\mapsto L such that A\otimes_K L is split; in the special case A=K[x]/f(x) this is the same as a splitting field of the polynomial f(x
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

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