"algebraic extension" meaning in All languages combined

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        {
          "ref": "1964, Shreeram Shankar Abhyankar, Local Analytic Geometry, Academic Press, page 200:",
          "text": "What we now have to prove is that: if an overring R#x2A; of R is such that R#x2A; is an integral domain, R#x2A; is integral over R and the quotient field #x5C;mathfrak#x7B;L#x7D;#x2A; of R#x2A; is a finite algebraic extension of #x5C;mathfrak#x7B;L#x7D;, then R#x2A; is a local ring and a finite R-module.",
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          "ref": "2005, Manuel Bronstein, Symbolic Integration I: Transcendental Functions, 2nd edition, Springer, page 110:",
          "text": "It turns out that when an irreducible polynomial splits in an algebraic extension, then the order at the new irreducible factors remains the same as before for arguments that are defined over the ground field.",
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          "ref": "2013, Askold Khovanskii, Galois Theory, Coverings, and Riemann Surfaces, Springer, page 65:",
          "text": "We describe the correspondence between subfields of the field P#x5F;a(O) that are algebraic extensions of the field K(X) and the subgroups of finite index of the fundamental group #x5C;pi#x5F;1(X#x5C;setminusO).",
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      "glosses": [
        "A field extension L/K which is algebraic over K (i.e., is such that every element of L is a root of some (nonzero) polynomial with coefficients in K)."
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          "word": "associative algebra"
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          "word": "Galois extension"
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        "(algebra, field theory) A field extension L/K which is algebraic over K (i.e., is such that every element of L is a root of some (nonzero) polynomial with coefficients in K)."
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      "word": "normal extension"
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      "word": "separable extension"
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          "text": "What we now have to prove is that: if an overring R#x2A; of R is such that R#x2A; is an integral domain, R#x2A; is integral over R and the quotient field #x5C;mathfrak#x7B;L#x7D;#x2A; of R#x2A; is a finite algebraic extension of #x5C;mathfrak#x7B;L#x7D;, then R#x2A; is a local ring and a finite R-module.",
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          "ref": "2005, Manuel Bronstein, Symbolic Integration I: Transcendental Functions, 2nd edition, Springer, page 110:",
          "text": "It turns out that when an irreducible polynomial splits in an algebraic extension, then the order at the new irreducible factors remains the same as before for arguments that are defined over the ground field.",
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        },
        {
          "ref": "2013, Askold Khovanskii, Galois Theory, Coverings, and Riemann Surfaces, Springer, page 65:",
          "text": "We describe the correspondence between subfields of the field P#x5F;a(O) that are algebraic extensions of the field K(X) and the subgroups of finite index of the fundamental group #x5C;pi#x5F;1(X#x5C;setminusO).",
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        "A field extension L/K which is algebraic over K (i.e., is such that every element of L is a root of some (nonzero) polynomial with coefficients in K)."
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