Separable polynomial: Difference between revisions
(Created page with '{{univariate polynomial upto associates property}} ==Definition== Let <math>K</math> be a field and <math>f(x) \in K[x]</math> be a nonzero polynomial. We say that <math>f</mat...') |
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# <math>f(x)</math> splits completely into ''distinct'' linear factors over its splitting field. | # <math>f(x)</math> splits completely into ''distinct'' linear factors over its splitting field. | ||
# For any field <math>L</math> containing <math>K</math>, <math>f</math> is a square-free polynomial over <math>L</math>, i.e., no square of a polynomial divides <math>f</math>. | # For any field <math>L</math> containing <math>K</math>, <math>f</math> is a square-free polynomial over <math>L</math>, i.e., no square of a polynomial divides <math>f</math>. | ||
# The [[defining ingredient::discriminant of a polynomial|discriminant]] of <math>f</math> is nonzero. | |||
Latest revision as of 23:29, 15 May 2009
Template:Univariate polynomial upto associates property
Definition
Let be a field and be a nonzero polynomial. We say that is a separable polynomial if the following equivalent conditions are satisfied:
![{\displaystyle f(x)\in K[x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3a793fcdeee093d3c86ca1126bb35aa5e000e2c)
- and its formal derivative are relatively prime in .
- splits completely into distinct linear factors over its splitting field.
- For any field containing , is a square-free polynomial over , i.e., no square of a polynomial divides .
- The discriminant of is nonzero.
![{\displaystyle K[x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a9e6c2ac2830d6a9abe078b47450777c41d69a9)
