## You are here

Homeexample of nonperfect field

## Primary tabs

# example of nonperfect field

In this entry, we exhibit an example of a field that is not a perfect field.

Let $F=\mathbb{F}_{p}(t)$, where $\mathbb{F}_{p}$ is the field with $p$ elements and $t$ transcendental over $\mathbb{F}_{p}$. The splitting field $E$ of the irreducible polynomial $f=x^{p}-t$ is not separable over $F$. Indeed, if $\alpha$ is an element of $E$ such that $\alpha^{p}=t$, we have

$x^{p}-t=x^{p}-\alpha^{p}=(x-\alpha)^{p},$ |

which shows that $f$ has one root of multiplicity $p$.

Major Section:

Reference

Type of Math Object:

Example

Parent:

## Mathematics Subject Classification

12F10*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