separable polynomial
English
Noun
separable polynomial (plural separable polynomials)
- (algebra, field theory) A polynomial over a given field that has distinct roots in the algebraic closure of said field (the number of roots being equal to the degree of the polynomial).
- Over a perfect field, the separable polynomials are precisely the square-free polynomials.
- The study of the automorphisms of splitting fields of separable polynomials over a field is referred to as Galois theory.
- 1978, Marvin Marcus, Introduction to Modern Algebra, M. Dekker, page 277,
- We know that is a normal extension because it is the splitting field of the separable polynomial (see Theorem 7.5).
- 2005, Arne Ledet, Brauer Type Embedding Problems, American Mathematical Society, page 6,
- Proposition 1.4.2 A finite field extension is Galois if and only if is the splitting field over of a separable polynomial.
- 2006, Philippe Gille, Tamás Szamuely, Central Simple Algebras and Galois Cohomology, Cambridge University Press, page 321,
- If is a separable polynomial in , then the derivative is prime to in , and therefore a unit in . […] In the case when is an inseparable polynomial we may write for a suitable and separable polynomial .
Coordinate terms
- inseparable polynomial
Derived terms
- discriminantly separable polynomial
Related terms
- separable extension
Translations
polynomial that has distinct roots
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See also
- splitting field
- square-free polynomial
Further reading
Separable extension on Wikipedia.Wikipedia Perfect field on Wikipedia.Wikipedia Square-free polynomial on Wikipedia.Wikipedia Multiplicity (mathematics)#Multiplicity of a root of a polynomial on Wikipedia.Wikipedia- Separable Polynomial on Wolfram MathWorld
- Separable extension on Encyclopedia of Mathematics
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