Freshman's dream

The identity \((a+b)^p=a^p+b^p\) holds in every commutative ring of prime characteristic \(p\). It is a special case of the Frobenius endomorphism, not a rule for arbitrary exponents or rings.

Binomial theorem: A formula expressing powers of a sum using binomial coefficients. Its expansion shows which intermediate terms must vanish for the identity to hold.
Commutative ring: A ring whose multiplication satisfies ab=ba for every pair of elements. The identity is stated for this setting, where the binomial theorem applies.
Freshman's dream modulo p: The congruence (a+b)^p\equiv a^p+b^p\pmod p for integers a and b. Reduction modulo a prime gives a direct integer-arithmetic form of the ring identity.
Characteristic zero: The condition that no positive integer multiple of a ring's identity is zero. In characteristic zero, the intermediate binomial terms do not disappear in general.
Frobenius endomorphism: The map sending each element of a ring of characteristic p to its pth power. The identity says this map preserves addition as well as multiplication.
Field: A commutative ring in which every nonzero element has a multiplicative inverse. Every field of characteristic p provides a familiar setting for the identity.
Polynomial factorization over finite fields: The decomposition of polynomials into irreducible factors over finite fields. Frobenius powers help test polynomial relations and organize finite-field computations.
Composite characteristic: A ring characteristic greater than one that is composite rather than prime. The prime-exponent argument does not apply directly to a composite characteristic.
Binomial coefficient: An integer counting the ways to choose a fixed-size subset from a finite set. The intermediate coefficients are multiples of p and therefore zero in characteristic p.
Finite field: A field with finitely many elements, necessarily having prime-power size. Finite fields of characteristic p exhibit the identity through their Frobenius maps.

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