What does the adjoining mean?

Question

What is the adjoin operation? The wikipedia link is pretty scant, but from eat it appears to be something along the lines as the smallest step towards the union of two sets? Are Adjunction and Union relatated?

Answer 1

There are two kinds of adjunction. The easiest is when your field $K$ and your element $\alpha\notin K$ are both sitting inside a bigger field $\Omega$. Then $K(\alpha)$ may be defined to be the smallest subfield of $\Omega$ that contains both $K$ and $\alpha$: it’s the intersection of all subfields $L\subset\Omega$ that themselves contain $K$ and $\alpha$, and there are other ways of describing it.

The other kind is sometimes called “abstract adjunction”, where you take a polynomial $f$ irreducible over $K$, and make up a new field $L$, defined to be $K[X]/(f(X))$, which contains a naturally isomorphic copy of $K$, and an element $\widetilde X$ that’s a root of $f$. This new field also has the property that nothing smaller than it contains both (the isomorphic copy of) $K$ and the root.

I don’t think that thinking of either of these processes as something like union would be at all helpful.