Freshman’s dream

The Freshman’s dream

(𝑥 + 𝑦) 𝑞 = 𝑥 𝑞 + 𝑦 𝑞

does not holds in $ℤ$ , but is true of $𝑞 = 𝑝 ℎ$ in any commutative ring of prime characteristic $𝑝$ , ring since applying the binomial expansion

(𝑥 + 𝑦) 𝑞 = 𝑞 \sum 𝑘 = 0 (𝑞 𝑘) 𝑥 𝑘 𝑦 𝑛 - 𝑘

and $𝑝 ∣ (𝑞 𝑘)$ for $0 < 𝑘 < 𝑛$ . Alternatively, one can do induction on the case where $𝑞 = 𝑝$ is a prime.