A field contains $ℤ 𝑝$ or $ℚ$

Let $𝐾$ be a field. Then $𝐾$ has a unique subfield isomorphic to modular arithmetic $ℤ 𝑝$ for some prime $𝑝$ or $ℚ$ . #m/thm/ring Thus $ℤ 𝑝$ and $ℚ$ are the only prime fields.

Proof

The characteristic of an integral domain is 0 or prime, and A ring contains $ℤ$ or $ℤ 𝑛$ where $𝑛 = c h a r (𝐾)$ . This covers the $ℤ 𝑝$ case. If $c h a r (𝐾) = 0$ , then let $𝐼 : ℤ ↪ 𝐾$ be the unique initial ring homomorphism and

𝑄 = {𝑎 𝑏 - 1 : 𝑎 𝑏 - 1 \in 𝐼 (ℤ), 𝑏 \neq 0}

Then $𝑄 ≅ ℚ$ .

Thus $𝖥 𝗅 𝖽 𝑝$ has $ℚ$ initial iff $𝑝 = 0$ and $ℤ 𝑝$ initial if $𝑝$ is prime.

#state/tidy | #lang/en | #SemBr

A field contains ℤ𝑝 or ℚ

A field contains $ℤ 𝑝$ or $ℚ$