See separable polynomial on Wiktionary
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"word": "discriminantly separable polynomial"
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"text": "Over a perfect field, the separable polynomials are precisely the square-free polynomials.",
"type": "example"
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"text": "The study of the automorphisms of splitting fields of separable polynomials over a field is referred to as Galois theory.",
"type": "example"
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{
"ref": "1978, Marvin Marcus, Introduction to Modern Algebra, M. Dekker, page 277:",
"text": "We know that F(#x5C;zeta) is a normal extension because it is the splitting field of the separable polynomial xⁿ-1 (see Theorem 7.5).",
"type": "quote"
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{
"ref": "2005, Arne Ledet, Brauer Type Embedding Problems, American Mathematical Society, page 6:",
"text": "Proposition 1.4.2 A finite field extension M#x2F;K is Galois if and only if M is the splitting field over K of a separable polynomial.",
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{
"ref": "2006, Philippe Gille, Tamás Szamuely, Central Simple Algebras and Galois Cohomology, Cambridge University Press, page 321:",
"text": "If #x5C;pi#x5F;P is a separable polynomial in K#x5B;t#x5D;, then the derivative #x5C;partial#x5F;t#x5C;pi#x5F;P is prime to #x5C;pi#x5F;P in K#x5B;t#x5D;, and therefore a unit in K#x5B;t#x5D;#x5F;P.[…]In the case when #x5C;pi#x5F;P is an inseparable polynomial we may write #x5C;pi#x5F;P#x3D;f(t#x7B;pʳ#x7D;) for a suitable r#x3E;0 and separable polynomial f.",
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"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)."
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"(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)."
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"sense": "polynomial that has distinct roots",
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"word": "polinomio separabile"
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"text": "Over a perfect field, the separable polynomials are precisely the square-free polynomials.",
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"text": "The study of the automorphisms of splitting fields of separable polynomials over a field is referred to as Galois theory.",
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"ref": "1978, Marvin Marcus, Introduction to Modern Algebra, M. Dekker, page 277:",
"text": "We know that F(#x5C;zeta) is a normal extension because it is the splitting field of the separable polynomial xⁿ-1 (see Theorem 7.5).",
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"ref": "2005, Arne Ledet, Brauer Type Embedding Problems, American Mathematical Society, page 6:",
"text": "Proposition 1.4.2 A finite field extension M#x2F;K is Galois if and only if M is the splitting field over K of a separable polynomial.",
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"ref": "2006, Philippe Gille, Tamás Szamuely, Central Simple Algebras and Galois Cohomology, Cambridge University Press, page 321:",
"text": "If #x5C;pi#x5F;P is a separable polynomial in K#x5B;t#x5D;, then the derivative #x5C;partial#x5F;t#x5C;pi#x5F;P is prime to #x5C;pi#x5F;P in K#x5B;t#x5D;, and therefore a unit in K#x5B;t#x5D;#x5F;P.[…]In the case when #x5C;pi#x5F;P is an inseparable polynomial we may write #x5C;pi#x5F;P#x3D;f(t#x7B;pʳ#x7D;) for a suitable r#x3E;0 and separable polynomial f.",
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{
"code": "it",
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"sense": "polynomial that has distinct roots",
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"word": "polinomio separabile"
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"code": "es",
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"sense": "polynomial that has distinct roots",
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"word": "separable polynomial"
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