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15
votes
6answers
1k views

In what sense are fields an algebraic theory?

Since there is no "free field generated by a set", it would seem that 1) there is no monad on Set whose algebras are exactly the fields and 2) there is no Lawvere theory whose models in Set are ...
44
votes
9answers
9k views

Galois Groups vs. Fundamental Groups

In a recent blog post Terry Tao mentions in passing that: "Class groups...are arithmetic analogues of the (abelianised) fundamental groups in topology, with Galois groups serving as the analogue of ...
-2
votes
1answer
576 views

Fundamental: Division by Zero [closed]

All the articles I've read regarding "Division by Zero" the main argument for it being an undefined operation, because all proofs lead to contradictions. ...
11
votes
3answers
4k views

Finite extension of fields with no primitive element

What is an example of a finite field extension which is not generated by a single element? Background: A finite field extension E of F is generated by a primitive element if and only if there are a ...