Freshman's dream

Builds on Commutative ring Characteristic of a ring

Statement

Let $p$ be a prime number and let $R$ be a commutative ring of characteristic $p$ . Then $(x + y)^p = x^p + y^p$ for all $x, y \in R$ .

Remarks

The proof uses the binomial theorem. For $0 < k < p$ , the prime $p$ divides the binomial coefficient $\binom{p}{k}$ , so these terms vanish in $R$ . By induction, $(x + y)^{p^n} = x^{p^n} + y^{p^n}$ for every $n \ge 0$ . The identity gives the Frobenius endomorphism. In $\mathbb{R}$ the identity is false: $(1 + 1)^2 = 4$ and $1^2 + 1^2 = 2$ .

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Group concept
Ring definition
Characteristic of a ring definition
Commutative ring definition
Freshman's dream statement

Where this leads

7 pages build on this one.

Tag	`0396VGFA78`
URI	`m2://0396VGFA78`
Kind	statement, theorem
Writing	W1 stated
MSC	13A35
Status	This copy of the site has no build data. Run `m2 status 0396VGFA78`.
Uses	Commutative ring, Characteristic of a ring
Same as	Wikidata Q4120528 Wikipedia Freshman's dream

Freshman's dream

Statement

Remarks

Reading path

Where this leads

Further reading

What this page needs for W2

What links here

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