Let $F$ be a field. . The $\it{prime}$ $\it{subfield}$ $\it{of}$ $F$ is the intersection of all subfields of $F$. Show that this subfield is the quocient field of the prime subrings of $F$, is contained inside all subfields of $F$, and is isomorphic to $\mathbb{F}_{p}$ or $\mathbb{Q}$ depending on whether the caracteristict of $F$ is $p > 0$ or $0$.

I can prove almost every affirmation, except the first. If necessary, I can write my proof of the others affirmations here. I would like any hint about the first. Thanks for the help.

  • $\begingroup$ A prime subring is the intersection of all subrings I suppose? Then just use the characterisation of the quotient field as the smallest field containing the ring and the fact that fields are also rings. $\endgroup$ – Verdruss Mar 20 '18 at 22:20
  • $\begingroup$ Maybe the prime subring is the image of $\Bbb Z$ under the natural map. $\endgroup$ – Lubin Mar 21 '18 at 5:29

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