Field

A field contains ℤ𝑝 or ℚ

Let 𝐾 be a field. Then 𝐾 has a unique subfield isomorphic to modular arithmetic ℤ𝑝 for some prime 𝑝 or Rational numbers. ring Thus ℤ𝑝 and ℚ are the only prime fields.

Proof

The characteristic of an integral domain is 0 or prime, and A ring contains the integers or modular arithmetic where 𝑛 =char(𝐾). This covers the ℤ𝑝 case. If char(𝐾) =0, then let 𝐼 :ℤ ↪𝐾 be the unique initial ring homomorphism and

𝑄={𝑎𝑏−1:𝑎𝑏−1∈𝐼(ℤ),𝑏≠0}

Then 𝑄 ≅ℚ.

Thus Category of fields of characteristic p has ℚ initial iff 𝑘 =0 and ℤ𝑝 initial if 𝑝 is prime.

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