1.10 Stem fields

Let $f$ be a monic irreducible polynomial in $F[X]$ . A pair $(E,\alpha)$ consisting of an extension $E$ of $F$ and an $\alpha\in E$ is called⁷⁷ 7 Following A.A. Albert (Modern Higher Algebra, 1937) who calls the splitting field of a polynomial its root field. a stem field for $f$ if $E=F[\alpha]$ and $f(\alpha)=0$ . For example, the pair $(E,\alpha)$ with $E=F[X]/(f)=F[x]$ and $\alpha=x$ is a stem field for $f$ . Let $(E,\alpha)$ be a stem field, and consider the surjective homomorphism of $F$ -algebras

g(X)\mapsto g(\alpha)\colon F[X]\rightarrow E\text{.}

Its kernel is generated by a nonzero monic polynomial, which divides $f$ , and so must equal it. Therefore the homomorphism defines an $F$ -isomorphism

x\mapsto\alpha\colon F[x]\rightarrow E,\quad\text{where }F[x]=F[X]/(f)\text{.}

In other words, the stem field $(E,\alpha)$ of $f$ is $F$ -isomorphic to the standard stem field $(F[X]/(f),x)$ . It follows that every element of a stem field $(E,\alpha)$ for $f$ can be written uniquely in the form

a_{0}+a_{1}\alpha+\cdots+a_{m-1}\alpha^{m-1},\quad a_{i}\in F,\quad m=\deg(f)% \text{,}

and that arithmetic in $F[\alpha]$ can be performed using the same rules as in $F[x]$ . If $(E^{\prime},\alpha^{\prime})$ is a second stem field for $f$ , then there is a unique $F$ -isomorphism $E\rightarrow E^{\prime}$ sending $\alpha$ to $\alpha^{\prime}$ . We sometimes abbreviate “stem field $(F[\alpha],\alpha)$ ” to “stem field $F[\alpha]$ ”.