In abstract algebra, field extensions are the main object of study in field theory.

In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring.

Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces.

Can NATO survive defeat in Afghanistan? The question is far from abstract.

The analysis of SDSs uses techniques from combinatorics, abstract algebra, graph theory, dynamical systems and probability theory.

And here we see Galois starting to develop a language to see the substance of the things unseen, the sort of abstract idea of the symmetry underlying this physical object.

In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative.

Now to put some neurobiological meat on this abstract model, we constructed a simple one dimensional world for our favorite subject, fruit flies.

The first step is to select material with abstract content to challenge students' logic, and that will therefore have different meaning to different students, in order to aid discussion.

And the power of Galois' mathematical language is it also allows us to create symmetrical objects in the unseen world, beyond the two dimensional, three dimensional, all the way through to the four or five or infinite dimensional space.

They all look very different. But this language allows us to say that they are representations of the same symmetrical abstract object, which we call 4 4 2. Nothing to do with football, but because of the fact that there are two places where you can rotate by a quarter of a turn, and one by half a turn.