algebraic extension

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algebraic extension \Prononciation ?\

1. (Algèbre) Extension algébrique.
   • What we now have to prove is that: if an overring ${\displaystyle R*}$ of ${\displaystyle R}$ is such that ${\displaystyle R*}$ is an integral domain, ${\displaystyle R*}$ is integral over ${\displaystyle R}$ and the quotient field ${\displaystyle {\mathfrak {L}}*}$ of ${\displaystyle R*}$ is a finite algebraic extension of ${\displaystyle {\mathfrak {L}}}$, then ${\displaystyle R*}$ is a local ring and a finite ${\displaystyle R}$-module. — (Shreeram Shankar Abhyankar, Local Analytic Geometry, Academic Press, 1964, page 200)

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   • 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. — (Manuel Bronstein, Symbolic Integration I: Transcendental Functions, Springer, 2005)

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• transcendental extension

• field extension

• normal extension
• separable extension
• Galois extension

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