Talk:root field

Definition

@AugPi Where did you get this definition from? I know the root field as the smallest extension where the polynomial has one root (i.e., is reducible), not where it decomposes into linear factors. 𝟙𝟤𝟯𝟺𝐪𝑤𝒆𝓇𝟷𝟮𝟥𝟜𝓺𝔴𝕖𝖗𝟰 (𝗍𝗮𝘭𝙠) 16:07, 30 April 2022 (UTC)

Garrett Birkhoff with Saunders Mac Lane (1953) A Survey Of Modern Algebra, Revised edition, U.S.A.: The Macmillan Company, published 1960, §XV.1, page 427: “DEFINITION. An extension N of F is a root field of a polynomial f(x) of degree n with coefficients in F if: (i) f(x) can be factored into linear factors $f(x)=c(x-u_{1})...(x-u_{n})$ in N, and (ii) N is generated over F by the roots of f(x), as $N=F(u_{1},...,u_{n})$.”

According to (i), the extension N splits the polynomial f(x) (or “f(x) splits over N”), and according to (ii), N is the smallest extension of F which splits f(x), ∴ the “root field” N qualifies as a splitting field.   AugPi 16:16, 6 May 2022 (UTC) @1234qwer1234qwer4