What is the difference between field theory and Galois theory I am about to finish the book Galois theory by Harold Edwards. I am planning to study Galois theory at a more advanced level or field theory. I am unable to decide because I don't know the difference between Galois theory and field theory.
Question: What is the difference between field theory and Galois theory ?
My guess is the Galois theory would be a "subset" of field theory, no ? If my guess is true I don't know whether Galois theory is a proper subset of field theory or not. If my guess is true, I'd like to have an idea about motivations about topics in field theory that do not fall under Galois theory.
Thank you
 A: Galois theory is the theory of the duality between profinite groups associated to fields and closed subgroups which arise as dual to field extensions of the original field. It's about the algebra of polynomials over a field and how that helps to understand other fields constructed algebraically from the original field, i.e. from roots of polynomials over a field. Some basic examples are
$$\Bbb Q(i)=\Bbb Q[x]/(x^2+1), \quad \Bbb Q[x]/(x^5+x-1)$$
The latter has no simple description in terms of the usual algebraic operations such as $\sqrt[n]\cdot$ or $+,-,\cdot, /$, but is nonetheless an example because it talks about roots of a polynomial with rational coefficients. It was actually the question of solving polynomials using radicals that originally motivated Galois to create the notion which we now call Galois theory.
which is constructed from the rationals by adding in the square root of $-1$, which is a root of a polynomial with coefficients in the base field, $\Bbb Q$.
You are correct in saying it is one aspect of studying fields, at least the classical theory is.
Due to the implications in how Galois groups act, it also has applications in understanding integer rings for global fields including ramification, inertia, and splitting behavior of primes (see Artin reciprocity for the abelian case) and in today's mathematical world the non-commutative Galois theory is connected to the Langlands program, a popular area of study related to things such as automorphic forms and other areas of number theory. For elliptic curves defined over a field, the Galois action is indispensable particularly for the representation on the $\ell$-adic Tate module. Other topics include Galois cohomology from which we derive the invaluable Hilbert Theorem 90 and Kümmer theory, both central to class field theory and the latter helping to describe a wide class of useful algebraic extensions, particularly for radical and abelian ones.
For topics that do not fall under the purvey of Galois theory, you are considering non-algebraic aspects of fields, eg. transcendence, which is the case when you totally lack a polynomial and your extension is of infinite degree, with no way to break it up via finite pieces. The other case is the inseparable one where your polynomials are in some sense "simple," but are too simple to give rise to the structure of the group symmetry because the roots can't be separated from one another.
An example of each is
$$\Bbb F_p(t)[x]/(x^p-t)$$
where the extension is no longer separable because all the roots of the polynomial are the same because of the positive characteristic, in fact, if $z$ is one such, we see $x^p-t=(x-z)^p$.
$$\Bbb{Q}(x_1,\ldots, x_n)/\Bbb Q$$
where the elements adjoined are a formal variables, not roots of a polynomial in $\Bbb Q$, so gives rise to an infinite (transcendental) extension. These extensions are quite useful in that they can help to prove that all finite groups are the Galois group for some field (not to be confused with the inverse Galois problem which deals with extensions only of $\Bbb Q$). This is also part of the crux of the idea that all profinite groups are Galois groups for some field extension.
