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Comment by revskill | Hacker News Reader

So what is the point of being a field ?

Over fields, polynomials mostly behave as expected, and systems of linear equations are solved very similarly to R. Basically, you can adapt quite a lot of real and complex algorithms to other fields, including matrix operations.

Once you leave fields and then even integral domains, things get weird. For example, the quadratic equation x^2 = 1 has four roots in Z_8.

You get "division".

The high level answer is that every module over a field is free. That is, if F is a field and M is an F-module then M is isomorphic to a direct sum of F, which may be a finite or infinite direct sum.

It’s an abstraction that helps mathematicians study interesting phenomena. I believe the famous squaring the circle problem was resolved using the language of fields.

That we can't square the circle comes from pi being transcendental. The result that you're thinking of is Galois' proof that there is no algebraic formula forroots of 5th degree polynomials.

"transcendental" is field language

I've always thought of "transcendental" as number theory language, though I can see how someone could argue that it is field language.

But the Galois group of a field extension definitely is field language.

a field extension is the thing which is transcendental or not.

Yeah, and constructability is usually handled by proving that a length is constructable if it lives in an iterated quadratic extension of the rationals. Pi does not lie in such an extension, so is not a constructable length (and neither is its square root).

Fields are easier to deal with.

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