Polynomials | Field (mathematics)

Separable polynomial

In mathematics, a polynomial P(X) over a given field K is separable if its roots are distinct in an algebraic closure of K, that is, the number of distinct roots is equal to the degree of the polynomial. This concept is closely related to square-free polynomial. If K is a perfect field then the two concepts coincide. In general, P(X) is separable if and only if it is square-free over any field that contains K,which holds if and only if P(X) is coprime to its formal derivative Dโ€‰P(X). (Wikipedia).

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๐Ÿ‘‰ Learn how to classify polynomials. A polynomial is an expression of the sums/differences of two or more terms having different interger exponents of the same variable. A polynomial can be classified in two ways: by the number of terms and by its degree. A monomial is an expression of 1

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From playlist Classify Polynomials

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From playlist Galois theory

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From playlist Calculus 1 Playlist 1

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From playlist Spring 2021 Online Kolchin Seminar in Differential Algebra

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From playlist Mathematics

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From playlist Classify Polynomials | Equations

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Classify a polynomial and determine degree and leading coefficient

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From playlist Classify Polynomials | Equations

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Classify a polynomial and determine degree and leading coefficient

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From playlist Classify Polynomials | Equations

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Classify a polynomial and determine degree and leading coefficient

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From playlist Classify Polynomials | Equations

Related pages

Galois theory | Rational function | Tensor product of fields | Algebraic closure | Coefficient | Finite field | Frobenius endomorphism | Automorphism | Cyclic permutation | Minimal polynomial (field theory) | Additive polynomial | Group (mathematics) | Permutation | Polynomial | Nilpotent | Mathematical proof | Alternating group | Rational number | Perfect field | Degree of a polynomial | Discriminant | Separable extension | Splitting field | Field extension | Projective line | Algebraic element | Characteristic (algebra) | Mathematics | Field (mathematics) | Integer | Formal derivative | Algebraic geometry | Ring (mathematics) | Square-free polynomial | Galois group | Prime number | Irreducible polynomial | Subgroup | Modular arithmetic | Commutative ring