# primitive element theorem

###### Theorem 1.

Let $F$ and $K$ be arbitrary fields, and let $K$ be an extension^{} of $F$ of finite degree. Then there exists an element $\alpha \mathrm{\in}K$ such that $K\mathrm{=}F\mathit{}\mathrm{(}\alpha \mathrm{)}$ if and only if there are finitely many fields $L$ with $F\mathrm{\subseteq}L\mathrm{\subseteq}K$.

Note that this implies that every finite separable extension^{} is not only finitely generated^{}, it is generated by a single element.

Let $X$ be an indeterminate^{}. Then $\mathbb{Q}(X,i)$ is not generated over $\mathbb{Q}$ by a single element (and there are infinitely many intermediate fields $\mathbb{Q}(X,i)/L/\mathbb{Q}$). To see this, suppose it is generated by an element $\alpha $. Then clearly $\alpha $ must be transcendental, or it would generate an extension of finite degree. But if $\alpha $ is transcendental, we know it is isomorphic^{} to $\mathbb{Q}(X)$, and this field is not isomorphic to $\mathbb{Q}(X,i)$: for example, the polynomial^{} ${Y}^{2}+1$ has no roots in the first but it has two roots in the second. It is also clear that it is not sufficient for every element of $K$ to be algebraic over $F$: we know that the algebraic closure^{} of $\mathbb{Q}$ has infinite degree over $\mathbb{Q}$, but if $\alpha $ is algebraic over $\mathbb{Q}$ then $[\mathbb{Q}(\alpha ):\mathbb{Q}]$ will be finite.

This theorem has the corollary:

###### Corollary 1.

Let $F$ be a field, and let $\mathrm{[}F\mathit{}\mathrm{(}\beta \mathrm{,}\gamma \mathrm{)}\mathrm{:}F\mathrm{]}$ be finite and separable. Then there exists $\alpha \mathrm{\in}F\mathit{}\mathrm{(}\beta \mathrm{,}\gamma \mathrm{)}$ such that $F\mathit{}\mathrm{(}\beta \mathrm{,}\gamma \mathrm{)}\mathrm{=}F\mathit{}\mathrm{(}\alpha \mathrm{)}$. In fact, we can always take $\alpha $ to be an $F$-linear combination^{} (http://planetmath.org/LinearCombination) of $\beta $ and $\gamma $.

To see this (in the case of characteristic^{} $0$), we need only show that there are finitely many intermediate fields. But any intermediate field is contained in the splitting field^{} of the minimal polynomials^{} of $\beta $ and $\gamma $, which is Galois with finite Galois group^{}. The explicit form of $\alpha $ comes from the proof of the theorem.

For more detail on this theorem and its proof see, for example, *Field and Galois Theory ^{}*, by Patrick Morandi (Springer Graduate Texts in Mathematics 167, 1996).

Title | primitive element theorem |
---|---|

Canonical name | PrimitiveElementTheorem |

Date of creation | 2013-03-22 11:45:48 |

Last modified on | 2013-03-22 11:45:48 |

Owner | alozano (2414) |

Last modified by | alozano (2414) |

Numerical id | 18 |

Author | alozano (2414) |

Entry type | Theorem |

Classification | msc 12F05 |

Classification | msc 65-01 |

Related topic | SimpleFieldExtension |

Related topic | PrimitiveElementOfBiquadraticField2 |

Related topic | PrimitiveElementOfBiquadraticField |