### Alex Nelson

Shared publicly -A field

I think the answer is, unsurprisingly,

Take this subring, and consider the field of fractions generated by this subring. Clearly the field of fractions is isomorphic to the rationals. (Sub-puzzle: is the isomorphism unique? Unique up to some equivalence?)

We need to show the field

(The embedding of the subring isomorphic to the integers should extend to the field of fractions, we just need to prove that its image on the field of fractions is actually contained in

**F**has characteristic*n*if the morphism of the integers into**F**generated by f(1)=1 has its smallest positive integer*n*be such that f(n)=0. (If no such*n*exists, for example the rationals has the integers as a subring, then we say it is characteristic 0.)**Puzzle:**Is there a field with characteristic 0 that does not have a subfield isomorphic to the rationals?I think the answer is, unsurprisingly,

*there is no such field*. If**F**has characteristic 0, then it has a subring isomorphic to the integers.Take this subring, and consider the field of fractions generated by this subring. Clearly the field of fractions is isomorphic to the rationals. (Sub-puzzle: is the isomorphism unique? Unique up to some equivalence?)

We need to show the field

**F**has the field of fractions as a subfield. Uh, that's left as an exercise for the reader...yeah ;)(The embedding of the subring isomorphic to the integers should extend to the field of fractions, we just need to prove that its image on the field of fractions is actually contained in

**F**. I honestly don't know off the top of my head if it will be contained in**F**or not!)1

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