Here's another unspoken rule of mathematics: English doesn't always commute!

Word order is important...

Today we've got a little post on a little fact relating to group actions. I wanted to write about this not so much to emphasize its importance (it's certainly not a major result) but simply to uncover the intuition behind it.

Today we're talking about Eisenstein's (not Einstein's!) Criterion - a well-known test to determine whether or not a polynomial is irreducible. In particular, we'll consider two examples where a not-so-obvious trick is needed in order to apply the criterion.

When I was first introduced to Noetherian rings, I didn't understand why my professor made such a big hoopla over these things. What makes Noetherian rings so special? Today's post is just a little intuition to stash in The Back Pocket, for anyone hearing the word "Noetherian" for the first time. 

You know the Sylow game. You're given a group of a certain order and are asked to show it's not simple. But where do you start? Here are four options that may be helpful when trying to produce a nontrivial normal subgroup.