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USAMO 1973 #4
*August 19, 2009*

*Posted by lumixedia in algebra, Problem-solving.*

Tags: algebra, contest math, olympiad math, USAMO, USAMO 1973

7 comments

Tags: algebra, contest math, olympiad math, USAMO, USAMO 1973

7 comments

A fairly straightforward algebra problem. Could appear on a modern AMC-12, though the decoy answers would have to be carefully written.

**USAMO 1973 #4.** Determine all the roots, real or complex, of the system of simultaneous equations

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Integrality, invariant theory for finite groups, and more tools for Noetherian testing
*August 11, 2009*

*Posted by Akhil Mathew in algebra, commutative algebra.*

Tags: algebra, integrality, invariant theory, Noetherian rings

6 comments

Tags: algebra, integrality, invariant theory, Noetherian rings

6 comments

There are quite a few more tools to tell whether a ring is Noetherian. In this post, I’ll discuss another basic tool: integrality. I’ll discuss the application to invariant theory for finite groups.

**Subrings **

In general, it is **not** true that a subring of a Noetherian ring is Noetherian. For instance, let be the polynomial ring in infinitely many variables over a field . Then is not Noetherian because of the ascending chain

However, the quotient field of is Noetherian. This applies to any non-Noetherian integral domain.

There are special cases where we can conclude a subring of a Noetherian ring is Noetherian.

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USAMO 1973 #2
*August 11, 2009*

*Posted by lumixedia in Problem-solving.*

Tags: algebra, contest math, number theory, olympiad math, USAMO, USAMO 1973

3 comments

Tags: algebra, contest math, number theory, olympiad math, USAMO, USAMO 1973

3 comments

**USAMO 1973 #2**. Let and denote two sequences of integers defined as follows:

Thus, the first few terms of the sequence are:

Prove that, except for “1”, there is no term which occurs in both sequences. (more…)

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“Undergraduate Algebra”: Or How I Relearned Algebra in a Week
*August 11, 2009*

*Posted by Martin Camacho in General.*

Tags: algebra, books

5 comments

Tags: algebra, books

5 comments

A few weeks ago I vowed to relearn all of my forgotten algebra – advanced group theory, rings, modules, and fields especially. The main problem,at least for me, was finding a viable resource to tutor me. Wikipedia proved futile as there was no use in clicking links in an unsystematic manner, and Wikibooks’ algebra section was simultaneously obtrusive and incomplete.

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How to tell if a ring is Noetherian
*August 9, 2009*

*Posted by Akhil Mathew in algebra, commutative algebra.*

Tags: algebra, commutative algebra, Hilbert basis theorem, localization, Noetherian rings

7 comments

Tags: algebra, commutative algebra, Hilbert basis theorem, localization, Noetherian rings

7 comments

I briefly outlined the definition and first properties of Noetherian rings and modules a while back. There are several useful and well-known criteria to tell whether a ring is Noetherian, as I will discuss in this post. Actually, I’ll only get to the first few basic ones here, though these alone give us a lot of tools for, say, algebraic geometry, when we want to show our schemes are relatively well-behaved. But there are plenty more to go.

**Hilbert’s basis theorem **

It is the following:

Theorem 1 (Hilbert)Let be a Noetherian ring. Then the polynomial ring is also Noetherian.

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Generic freeness II
*July 30, 2009*

*Posted by Akhil Mathew in algebra, commutative algebra.*

Tags: algebra, commutative algebra, generic freeness, localization

11 comments

Tags: algebra, commutative algebra, generic freeness, localization

11 comments

Today’s goal is to partially finish the proof of the generic freeness lemma; the more general case, with finitely generated algebras, will have to wait for a later time though.

Recall that our goal was the following:

Theorem 1Let be a Noetherian integral domain, a finitely generated -module. Then there there exists with a free -module.

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USAMO 1972 #4
*July 26, 2009*

*Posted by lumixedia in Problem-solving.*

Tags: algebra, contest math, olympiad math, USAMO, USAMO 1972

3 comments

Tags: algebra, contest math, olympiad math, USAMO, USAMO 1972

3 comments

**USAMO 1972 #4.** Let denote a non-negative rational number. Determine a fixed set of integers , , , , , such that, for *every* choice of ,

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The enveloping algebra
*July 25, 2009*

*Posted by Akhil Mathew in algebra.*

Tags: algebra, enveloping algebras, Lie algebras, tensor algebras, universal properties

1 comment so far

Tags: algebra, enveloping algebras, Lie algebras, tensor algebras, universal properties

1 comment so far

As we saw in the first post, a representation of a finite group can be thought of simply as a module over a certain ring: the group ring. The analog for Lie algebras is the enveloping algebra. That’s the topic of this post.

** Definition **

The basic idea is as follows. Just as a representation of a finite group was a group-homomorphism for a vector space, a representation of a Lie algebra is a Lie-algebra homomorphism . Now, is the Lie algebra constructed from an associative algebra, —just as is the group constructed from taking invertible elements.

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Engel’s Theorem and Nilpotent Lie Algebras
*July 23, 2009*

*Posted by Akhil Mathew in algebra, representation theory.*

Tags: algebra, Engel's theorem, Lie algebras, linear algebra, nilpotent

1 comment so far

Tags: algebra, Engel's theorem, Lie algebras, linear algebra, nilpotent

1 comment so far

Now that I’ve discussed some of the basic definitions in the theory of Lie algebras, it’s time to look at specific subclasses: nilpotent, solvable, and eventually semisimple Lie algebras. Today, I want to focus on nilpotence and its applications.

** Engel’s Theorem **

To start with, choose a Lie algebra for some finite-dimensional -vector space ; recall that is the Lie algebra of linear transformations with the bracket . The previous definition was in terms of matrices, but here it is more natural to think in terms of linear transformations without initially fixing a basis.

Engel’s theorem is somewhat similar in its statement to the fact that commuting diagonalizable operators can be simultaneously diagonalized.

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Why simple modules are often finite-dimensional II
*July 22, 2009*

*Posted by Akhil Mathew in algebra, representation theory.*

Tags: algebra, finite-dimensional vector spaces, Nakayama's lemma, representation theory, simple modules

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Tags: algebra, finite-dimensional vector spaces, Nakayama's lemma, representation theory, simple modules

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I had a post a few days back on why simple representations of algebras over a field which are finitely generated over their centers are always finite-dimensional, where I covered some of the basic ideas, without actually finishing the proof; that is the purpose of this post.

So, let’s review the notation: is our ground field, which we no longer assume algebraically closed (thanks to a comment in the previous post), is a -algebra, its center. We assume is a finitely generated ring over , so in particular Noetherian: each ideal of is finitely generated.

Theorem 1 (Dixmier, Quillen)If is a finite -module, then any simple -module is a finite-dimensional -vector space.