The Two Ideals of Fields

https://susam.net/two-ideals-of-fields.html

by susam • 1 year ago

58 50 1 year ago

mtsolitary3• 1 year ago

This is a great example of one of those things in abstract maths that is hard to follow when you learn it, but once you’ve been through it a few times and learnt the definitions to heart it’s really just a rephrasing of the definitions.

tux3• 1 year ago

Trivial in math is a term that refers to anything you've already learned.

You sometimes hear people say that math is tautological. But regardless of whether it's all just an elaborate rephrasing of the axioms, it's quite beautiful.

math_dandy• 1 year ago

Historically, mathematicians have spent a huge amount of time and effort formulating optimal axioms and foundations so that theorems would follow naturally from structure. Theorems following “trivially” from a theoretical framework that took years to develop isn’t an indictment of the theorem, but an endorsement of the incredible effort expended to develop an optimal context for expressing and understanding the theorem.

aleph_minus_one2• 1 year ago

Trivial in math is a term that refers to anything you've already learned.

According to a professor, "trivial" means: "If this is not trivial for you, you should see this as a clear signal that you should take this course seriously instead of slacking of, or even that you simply are in the wrong course."

gosub1002• 1 year ago

My pet peeve math term is "clear". A long time ago I thought could teach myself group theory by buying the Springer group theory book and reading it from chapter 1, 1 page at a time. But I was blocked within the first 5 pages because the axioms and first few proofs kept saying how "clear" it was that all the results followed. Unfortunately, it was not "clear" to me :(

xelxebar3• 1 year ago

Indeed. Famously, though, figuring out if something is a tautology is undecidable!

zem• 1 year ago

ironically the article itself uses "trivial" in its other, more mathematical sense :)

downboots• 1 year ago

How to distinguish mastery of a complex subject from parsing one formally expressed in a complex way?

aleph_minus_one2• 1 year ago

How to distinguish mastery of a complex subject from parsing one formally expressed in a complex way?

In my opinion: the difference between a complex subject and one formally expressed in a complex way is that in the former, the results that you get are really deep (understanding them at the end feels like a spiritual experience).

mtsolitary3• 1 year ago

I think what you call “parsing” is largely indistinguishable from mastery in a lot of fields, particularly in abstract mathematics

bubblyworld• 1 year ago

Yeah, as an example this is a simple corollary of the following result which gets used a lot: the quotient of a (unital, commutative) ring by a maximal ideal is always a field.

If your ring has only two ideals then the trivial ideal is maximal, and thus your ring is already a field!

The more you know, the more "shortcuts" you start seeing, I guess.

srean• 1 year ago

Spivak states in his "Calculus on Manifolds' that definitions should be hard (to refine and state) and when done well, the theorems easy.

almostgotcaught3• 1 year ago

this is a standard thing in "mature" areas of math and it's absolutely the opposite of what's good for the student (all of the machinery being hidden in the definition instead of developed in the theorem's proof).

EDIT: if you hate "a monad is a monoid in the category of endofunctors" then you also hate "definitions should be hard and theorems easy".

VladVladikoff2• 1 year ago

Is there a good place to start with this stuff? I’ve done undergrad engineering but this math is alien to me.

xelxebar3• 1 year ago

It depends a bit on your tastes. The content falls under the broad umbrella of Abstract Algebra, more specifically Ring Theory, or perhaps Field Theory if you squint a bit. Those are your keywords.

"Applications of Abstract Algebra with Maple and MATLAB" by Klinger, Sigmon, and Stitzinger is apparently good for those with an engineering background: https://www.maplesoft.com/books/details.aspx?id=624.

If you're committed, then any introductory text on abstract algebra or group theory might capture your interest.

I would recommend starting with applications or something as close to your wheelhouse as possible just to stay motivated. Abstract algebra, in particular, is known for requiring quite a lot of machinery before obviously connecting with other things, which can feel like an onslaught unless you're inherently interested.

Have fun though! It's really one of the deepest subjects in modern math, IMHO. Almost every field has been affected by it's results.

kevinventullo4• 1 year ago

FYI many sources do not count the entire ring as an ideal. If you do, you’d have to define “maximal ideal” to mean “an ideal that is maximal with respect to inclusion, ignoring the entire ring ideal.”

vouaobrasil2• 1 year ago

Pretty much most sources, actually. When actually working with ideals, there's almost never an advantage to consider the whole ring an ideal. So all ideals in virtually all the literature I've ever read were taken to be proper, i.e. proper subsets

WCSTombs• 1 year ago

I very strongly disagree. Without considering the whole ring to be an ideal, you can't even define "the ideal generated by some elements" because there may not be such an ideal, since it could be the whole ring. Likewise, you can't perform common operations on ideals like sums because they could result in the whole ring. In fact, I would say there is almost no advantage in excluding the ring itself from the set of ideals.

Just to check, I have three math textbooks from my college days that include the definition of an ideal, and none of them attempt to exclude the ring itself from the definition.

The obvious compromise is to introduce the concept of a proper ideal as an ideal that is a proper subset, and to use that when you need to exclude the ring itself. E.g., a maximal ideal is a proper ideal that is maximal with respect to inclusion.

Sniffnoy• 1 year ago

That's not my experience at all as a mathematician. "Maximal ideal" implicitly means "maximal proper ideal", yes, but generally ideals don't have to be proper unless specified so.

If you don't include the whole ring as an ideal, you can't even define ideal addition, etc. I took an algebra class once from a professor who decided to define "ideal" to mean "proper ideal". After a few weeks he had to give it up because it just became too much trouble for reasons like that; he had to too often say "possibly improper ideal", i.e., this convention had the opposite effect he intended! I can't think of any other source I've seen use that convention.

tome• 1 year ago

See also "too simple to be simple": https://ncatlab.org/nlab/show/too+simple+to+be+simple

mathgradthrow7• 1 year ago

Those sources are crazy. The ideals form a lattice under intersection and "+".

All of algebraic geometry (a very popular branch of mathematics for studying rings) is built on the lattice of ideals. There is no way of giving a ring this structure without the ring itself being the top element of this lattice.

What you probably mean to say is that there are sources that do not treat R as a prime ideal of itself.

kevinventullo4• 1 year ago

Yes, I was misremembering. My mistake!

vouaobrasil2• 1 year ago

Another interesting fact about fields is that a commutative ring is a field if and only if every ideal is a prime ideal. (Obviously, every ideal of a field is a prime ideal. The converse is more interesting...)

shiandow• 1 year ago

This is one of those cases where the algebraic geometry perspective is actually helpful (without too much trouble).

If the only subspaces are single points then the space is itself a point.

The Two Ideals of Fields

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