freshman’s dream
Theorem 1 (Freshman’s dream).
If if a field of characteristic (http://planetmath.org/Characteristic) (so is prime) then for all we have
Therefore is a field monomorphism (called a Frobenius monomorphism.)
When is finite then it is indeed an automorphism. A field is called a perfect field when the map is surjective.
The theorem is so named because it is a common mistake for freshman math students to make over the real numbers. However, as the characteristic of the real numbers is 0, this does not apply in any interesting way to that setting.
It should also be noted that the result applies only to powers of the characteristic, and not all exponents.
Proof.
The proof is an application of the binomial theorem. We prove it for first.
Now observe
As is prime and it follows and do not divide . As the field has characteristic , is an integer where
Thus .
Title | freshman’s dream |
---|---|
Canonical name | FreshmansDream |
Date of creation | 2013-03-22 15:51:17 |
Last modified on | 2013-03-22 15:51:17 |
Owner | Algeboy (12884) |
Last modified by | Algeboy (12884) |
Numerical id | 18 |
Author | Algeboy (12884) |
Entry type | Theorem |
Classification | msc 11T23 |
Classification | msc 11T30 |
Synonym | Frobenius Automorphism |
Related topic | PolynomialCongruence |