English Dictionary

algebraic extension

名詞

(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).

日本語の意味

体Kのすべての元が、K係数の非零多項式の根となるような、体L/Kの拡大を意味し、代数的拡大または代数拡大とも呼ばれる。 / ある体Kの拡大体Lにおいて、Lの各要素がK上の非零多項式の根として表される性質を持った拡大を指す。

To prove the theorem, we assume L is an algebraic extension of K and examine the minimal polynomials of its elements.

定理を証明するために、LをK上の代数拡大（すなわちLの任意の元がKの係数をもつ零でない多項式を満たすような体の拡大）と仮定し、その元の最小多項式を調べます。

意味（1）

(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).

What we now have to prove is that: if an overring R* of R is such that R* is an integral domain, R* is integral over R and the quotient field 𝔏* of R* is a finite algebraic extension of 𝔏, then R* is a local ring and a finite R-module.

algebraic extensions （ plural ）