PlanetMath: freshman's dream

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freshman's dream

(Theorem)

Theorem 1 (Freshman's dream) If if a field of characteristic (so is prime) then for all $x,y\in k$ we have

$\displaystyle (x+y)^{p^i}=x^{p^i}+y^{p^i}.$

Therefore $x\mapsto x^{p^i}$ 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.

$\displaystyle (x+y)^{p}=\sum_{i=0}^p \binom{p}{i} x^{i} y^{p-i}.$

Now observe

$\displaystyle \binom{p}{i}=\frac{p!}{(p-i)!i!}=p\!\cdot\!\frac{(p-1)!}{(p-i)! i!}.$

is prime and $1\leq i\leq p-1$ it follows

and

do not divide

. As the field

has characteristic

, $\frac{(p-1)!}{(p-i)!i!}$ is an integer

where

$\displaystyle \binom{p}{i}=pm\equiv 0 .$

Thus

Now for simply use induction:

$\displaystyle (x+y)^{p^i}=((x+y)^p)^{p^{i-1}}=(x^p+y^p)^{p^{i-1}} =x^{p^i}+y^{p^i}.$

$\qedsymbol$

"freshman's dream" is owned by Algeboy. [ full author list (3) ]

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See Also: polynomial congruence

Other names:

Frobenius Automorphism

Keywords:

Field automorphism, finite characteristic, positive characteristic

Attachments:: freshman's dream error (Example) by Wkbj79

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Cross-references: induction, integer, divide, binomial theorem, application, proof, exponents, characteristic, real numbers, surjective, map, perfect field, finite, monomorphism, field monomorphism, prime, field
There are 3 references to this entry.

This is version 15 of freshman's dream, born on 2006-04-17, modified 2007-05-18.
Object id is 7839, canonical name is FreshmansDream.
Accessed 3424 times total.

Classification:

AMS MSC:	11T30 (Number theory :: Finite fields and commutative rings :: Structure theory)
	11T23 (Number theory :: Finite fields and commutative rings :: Exponential sums)

Pending Errata and Addenda

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