# lattice of fields

Let $K$ be a field and $\overline{K}$ be its algebraic closure^{}. The set $\mathrm{Latt}(K)$ of all intermediate fields $E$ (where $K\subseteq E\subseteq \overline{K}$), ordered by set theoretic inclusion, is a poset. Furthermore, it is a complete lattice^{}, where $K$ is the bottom and $\overline{K}$ is the top.

This is the direct result of the fact that any topped intersection structure is a complete lattice, and $\mathrm{Latt}(K)$ is such a structure^{}. However, it can be easily proved directly: for any collection^{} of intermediate fields $\{{E}_{i}\mid i\in I\}$, the intersection^{} is clearly an intermediate field, and is the infimum^{} of the collection. The compositum of these fields, which is the smallest intermediate field $E$ such that ${E}_{i}\subseteq E$, is the supremum^{} of the collection.

It is not hard to see that $\mathrm{Latt}(K)$ is an algebraic lattice, since the union of any directed family of intermediate fields between $K$ and $\overline{K}$ is an intermediate field. The compact elements in $\mathrm{Latt}(K)$ are the finite algebraic extensions^{} of $K$. The set of all compact elements in $\mathrm{Latt}(K)$, denoted by ${\mathrm{Latt}}_{F}(K)$, is a lattice ideal, for any subfield^{} of a finite algebraic extension of $K$ is finite algebraic over $K$. However, ${\mathrm{Latt}}_{F}(K)$, as a sublattice, is usually not complete^{} (take the compositum of all simple extensions $\mathbb{Q}(\sqrt{p})$, where $p\in \mathbb{Z}$ are rational primes).

Title | lattice of fields |
---|---|

Canonical name | LatticeOfFields |

Date of creation | 2013-03-22 17:13:26 |

Last modified on | 2013-03-22 17:13:26 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 5 |

Author | CWoo (3771) |

Entry type | Definition |

Classification | msc 12F99 |