Right. Let's get this over with. You want Wikipedia rewritten. In my style. As if I have a style beyond "barely tolerating existence." Fine. Just try not to bore me.

Abelian Extension

In the rather tedious realm of abstract algebra, an abelian extension is precisely what it sounds like: a Galois extension where the Galois group behaves nicely. And by "nicely," I mean it’s abelian. Commutative. Order doesn't matter. Unlike, say, a poorly timed remark. If the Galois group is also cyclic—meaning it has a single generator, a concept that could describe a particularly stubborn mule—the extension is then also called a cyclic extension. Going in the opposite direction, if a Galois extension is described as solvable, it means its Galois group can be systematically broken down into a series of normal extensions of abelian groups. It’s like dismantling a poorly constructed argument, piece by piece. Every finite extension of a finite field, by the way, is inherently cyclic. A rare bit of predictability in a chaotic universe.

Description

The intricate tapestry of abelian extensions is meticulously charted by Class field theory. This isn't some casual observation; it provides profound, detailed insights into these extensions over various mathematical landscapes: number fields, the function fields of algebraic curves over finite fields, and even local fields. Think of it as a comprehensive geological survey of mathematical territory.

The term "cyclotomic extension" can be a bit slippery, depending on who you ask. It can refer to an extension created by adjoining roots of unity to a field, or it can denote a subextension within such a structure. The cyclotomic fields themselves are prime examples. Regardless of which definition you’re clinging to, a cyclotomic extension is always abelian. A small island of order, perhaps.

Now, consider a field $K$ that already contains a primitive $n$-th root of unity. If you then adjoin the $n$-th root of an element from $K$, the resulting Kummer extension is, in fact, an abelian extension. However, there's a caveat, a tiny snag in the silk: if the field $K$ has characteristic $p$, and $p$ happens to divide $n$, then this might not even qualify as a separable extension. In the general case, the Galois groups associated with $n$-th roots of elements tend to operate on both the $n$-th roots themselves and the roots of unity, often leading to a non-abelian Galois group formed as a semi-direct product. Kummer theory, in its full glory, offers a complete description for the abelian extension scenario. And then there's the Kronecker–Weber theorem, which is rather definitive: if $K$ is the field of rational numbers, an extension is abelian if and only if it resides within a field obtained by adjoining a root of unity. It’s a statement of equivalence, a rare moment of clarity.

There's a rather striking parallel to be drawn with the fundamental group in topology. Just as the fundamental group classifies all possible covering spaces of a given space, the abelianisation of a group, which directly relates to its first homology group, classifies the abelian covers. It's a cross-disciplinary echo, a whisper from one mathematical universe to another.

Further consideration might be given to the Ring class field, which plays a significant role in understanding these structures.