In field theory, the primitive element theorem states that every finite separable field extension is simple, i.e. generated by a single element. This theorem implies in particular that all algebraic number fields over the rational numbers, and all extensions in which both fields are finite, are simple.

Terminology

Let ${\displaystyle E/F}$ be a field extension. An element ${\displaystyle \alpha \in E}$ is a primitive element for ${\displaystyle E/F}$ if ${\displaystyle E=F(\alpha ),}$ i.e. if every element of ${\displaystyle E}$ can be written as a rational function in ${\displaystyle \alpha }$ with coefficients in ${\displaystyle F}$. If there exists such a primitive element, then ${\displaystyle E/F}$ is referred to as a simple extension.

If the field extension ${\displaystyle E/F}$ has primitive element ${\displaystyle \alpha }$ and is of finite degree ${\displaystyle n=}$, then every element ${\displaystyle \gamma \in E}$ can be written in the form

${\displaystyle \gamma =a_{0}+a_{1}{\alpha }+\cdots +a_{n-1}{\alpha }^{n-1},}$

for unique coefficients ${\displaystyle a_{0},a_{1},\ldots ,a_{n-1}\in F}$. That is, the set

${\displaystyle \{1,\alpha ,\ldots ,{\alpha }^{n-1}\}}$

is a basis for E as a vector space over F. The degree n is equal to the degree of the irreducible polynomial of α over F, the unique monic ${\displaystyle f(X)\in F}$ of minimal degree with α as a root (a linear dependency of ${\displaystyle \{1,\alpha ,\ldots ,\alpha ^{n-1},\alpha ^{n}\}}$).

If L is a splitting field of ${\displaystyle f(X)}$ containing its n distinct roots ${\displaystyle \alpha _{1},\ldots ,\alpha _{n}}$, then there are n field embeddings ${\displaystyle \sigma _{i}:F(\alpha )\hookrightarrow L}$ defined by ${\displaystyle \sigma _{i}(\alpha )=\alpha _{i}}$ and ${\displaystyle \sigma (a)=a}$ for ${\displaystyle a\in F}$, and these extend to automorphisms of L in the Galois group, ${\displaystyle \sigma _{1},\ldots ,\sigma _{n}\in \mathrm {Gal} (L/F)}$. Indeed, for an extension field with ${\displaystyle =n}$, an element ${\displaystyle \alpha }$ is a primitive element if and only if ${\displaystyle \alpha }$ has n distinct conjugates ${\displaystyle \sigma _{1}(\alpha ),\ldots ,\sigma _{n}(\alpha )}$ in some splitting field ${\displaystyle L\supseteq E}$.

Example

If one adjoins to the rational numbers ${\displaystyle F=\mathbb {Q} }$ the two irrational numbers ${\displaystyle {\sqrt {2}}}$ and ${\displaystyle {\sqrt {3}}}$ to get the extension field ${\displaystyle E=\mathbb {Q} ({\sqrt {2}},{\sqrt {3}})}$ of degree 4, one can show this extension is simple, meaning ${\displaystyle E=\mathbb {Q} (\alpha )}$ for a single ${\displaystyle \alpha \in E}$. Taking ${\displaystyle \alpha ={\sqrt {2}}+{\sqrt {3}}}$, the powers 1, α, α², α³ can be expanded as linear combinations of 1, ${\displaystyle {\sqrt {2}}}$, ${\displaystyle {\sqrt {3}}}$, ${\displaystyle {\sqrt {6}}}$ with integer coefficients. One can solve this system of linear equations for ${\displaystyle {\sqrt {2}}}$ and ${\displaystyle {\sqrt {3}}}$ over ${\displaystyle \mathbb {Q} (\alpha )}$, to obtain ${\displaystyle {\sqrt {2}}={\tfrac {1}{2}}(\alpha ^{3}-9\alpha )}$ and ${\displaystyle \sqrt{3}=-<!--\; -\; -->\frac{1}{2}({\mathrm{\alpha <!--\; \alpha \; -->}}^3-<!--\; -\; -->11\mathrm{\alpha <!--\; \alpha \; -->})}$Zdroj:https://en.wikipedia.org?pojem=Primitive_element_theorem
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