splitting field
Frequency: Zipf frequency 3.76 in modern English
Meaning
noun
(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.
noun
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).
noun
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.
Example Sentences
- "Theorem 3.2. If K is a field and f#92;inK#91;x#93; has degree n#92;ge 1, then there exists a splitting field F of f with #91;F#58;K#93;#92;len#33;"
- "In the case of the polynomial t²#43;1#92;in#92;R#91;t#93;, the situation is quite clear; its splitting field is #92;C since t²#43;1#61;(t#43;i)(t-i) where i#61;#92;sqrt#123;-1#125;."
- "Example 4.3. By definition, #92;Q(i) is the splitting field of x²#43;1 over #92;Q, and #92;C is the splitting field for the same polynomial, over #92;R. Example 4.4. The splitting field F of x⁸-1 over #92;Q is generated by #92;zeta#58;#61;2#123;2#92;pii#47;8#125;; indeed, the roots of x⁸-1 are all the 8-th roots of 1, and all of them are powers of #92;zeta:[…]In fact, #92;zeta is a root of the polynomial x⁴#43;1, which is irreducible over #92;Q; therefore F#61;#92;Q(#92;zeta) is 'already' the splitting field of x⁴#43;1."
- "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#92;mapstoL such that A#92;otimes#95;KL is split; in the special case A#61;K#91;x#93;#47;f(x) this is the same as a splitting field of the polynomial f(x)."
- "2001, T. Y. Lam, A First Course in Noncommutative Rings, Springer, 2nd Edition, page 117, Ex. 7.6. For a finite-dimensional k-algebra R, let T(R)= operatorname radR+[R,R], where [R,R] denotes the subgroup of R generated by ab-ba for all a,b∈R. Assume that k has characteristic p>0. Show that T(R)⊆a∈R:aforsomem>1, with equality if k is a splitting field for R."
- "Group algebras are defined over the prime field #92;Q or #92;mathbbF#95;p (depending on the characteristic), and by what we have just proved #92;QG and #92;mathbbF#95;pG have splitting fields that are finite degree extensions of the prime field.[…] Some other basic facts about splitting fields are left to the exercises at the end of this chapter. Thus, if A is a finite-dimensional algebra over a field F that is a splitting field for A and E#92;supsetF is a field extension, it is the case that every simple E#92;otimes#95;FA-module can be written in F (Exercises 4 and 8)."