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I am taking an introductory number theory course this term, and I have found that while my algebra has been for the most part sufficient, I am severely lacking in even the basics concerning fields and field extensions.

My question is, what are the basic concepts/fundamental theorems in the theory of fields and field extensions? For group theory, for instance, I would say that the first isomorphism theorem, the class equation, the Sylow theorems, and a healthy familiarity with the likes of $Z/nZ$ and $S_n$ encapsulate the top parts of what one might expect to learn in a one semester introduction to algebra (by which I don't mean that this is all that one would learn, but that a lot of the other things done would be done to support these top level concepts). Is there a similar list of concepts which would similarly encapsulate some sort of top level knowledge of the subject of fields and field extensions? Any good introductions to the subject are also much appreciated!

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This is a tall order, and any list such as you’re requesting will surely be at best very incomplete.

Some things you should know and understand: definition of a (commutative) field, and the knowledge that any ring homomorphism from one field to another is necessarily an injection; dimension of a vector space, and the notion of field extension degree $[L:K]$; the fact that if $k$ is a field, the polynomial ring $k[x]$ is a Principal Ideal Domain, and that the maximal ideals of this ring are exactly those generated by irreducible polynomials; the fact that if $f(x)$ is $k$-irreducible, then $k[x]/(f)$ is a field, within which $k$ may be considered a subfield, with the degree of the extension equaling the degree of $f$; the concepts of algebraic and transcendental field extensions; the concepts of the field generated by a set of elements in some larger field and the compositum of several fields contained in a larger field; the concepts of separable and radicial (purely inseparable) extensions, as well as the concepts of normal and Galois extensions; the fact that a finite separable extension may be generated by a single element; then all the results of Galois Theory, which I think you can fill in for yourself.

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I can't really make such a list, but I can point out some features which inform much of field theory:

In any algebraic structure, one of the primary objects of interest are the "structure-preserving maps" (homomorphisms, usually). Well, with fields, the only "field homomorphisms" are injective, so we are left studying basically "two kinds": inclusions, and automorphisms.

An inclusion gives the "larger" field (the extension) the structure of a vector space over the smaller field. This allows us to use the techniques of linear algebra in investigating fields.

With automorphisms, we can investigate the group structure of the various automorphism groups. This gives a connection between group theory, and field theory.

Finally, it turns out one of the most profitable ways to study a field $F$ is to "enlarge it too much", by creating the ring $F[x]$, and then taking the quotient ring via a maximal ideal (these correspond to "prime polynomials", and this works in much the same way as taking $\Bbb Z/p\Bbb Z$ creates a finite field).

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