## You are here

Homesplitting field

## Primary tabs

# splitting field

Let $f\in F[x]$ be a polynomial over a field $F$. A splitting field for $f$ is a field extension $K$ of $F$ such that

1. 2. $K$ is the smallest field with this property (any sub-extension field of $K$ which satisfies the first property is equal to $K$).

Theorem: Any polynomial over any field has a splitting field, and any two such splitting fields are isomorphic. A splitting field is always a normal extension of the ground field.

Related:

NormalExtension

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

12F05*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections

## Comments

## State of eucation

A few months ago I had been to my alma mater, a college in Mumbai, mainly to to meet one of the faculty members who is well-versed in Latex. He introduced me to a colleague. I was surprised to find that, although she teaches linear algebra or calculus she is totally ignorant of number theory; she had not heard of Fermat's theorem or Euler's totient. Q: Does such a situation prevail in USA?

Devaraj