freshman’s dream

Theorem 1 (Freshman’s dream).

If k if a field of characteristic ( p>0 (so p is prime) then for all x,yk we have


Therefore xxpi is a field monomorphism (called a Frobenius monomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath.)

When k is finite then it is indeed an automorphism. A field k is called a perfect fieldMathworldPlanetmath when the map is surjectivePlanetmathPlanetmath.

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.


The proof is an application of the binomial theorem. We prove it for p first.


Now observe


As p is prime and 1ip-1 it follows i! and (p-i)! do not divide p. As the field k has characteristic p, (p-1)!(p-i)!i! is an integer m where


Thus (x+y)p=xp+yp.

Now for pi simply use inductionMathworldPlanetmath:


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 AutomorphismMathworldPlanetmathPlanetmathPlanetmath
Related topic PolynomialCongruence