Number theory textbook based on the absolute Galois group? I've just finished reading Ash and Gross's Fearless Symmetry, a wonderful little pop mathematics book on, among other things, Galois representations.  The book made clear a very interesting perspective that I wasn't aware of before: that a large chunk of number theory can be thought of as a quest to understand $G = \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$.  For example, part of the reason to study elliptic curves is to describe two-dimensional representations of $G$, and reciprocity laws are secretly about ways to describe the traces of Frobenius elements in various representations.  (That's awesome!  Why didn't anybody tell me that before?)
Are there number theory textbooks (presumably not introductory, but hopefully not too sophisticated either) which explicitly take this as a guiding principle?  I think this is a great idea to organize things like quadratic reciprocity around and I'm wondering if anybody has decided to actually do that at the undergraduate (or introductory graduate, maybe) level.
Edit:  In response to some comments and at least one downvote, most of the other questions on MO about the absolute Galois group that I can find are about the state of the art, and the references in them seem fairly sophisticated.  But it seems to me there are still interesting things to say along the lines of Fearless Symmetry, but directed to an undergraduate or introductory graduate-level audience as a kind of "second course in number theory."  I'm imagining a textbook like Serre's Course in Arithmetic.
 A: In the 1-dimensional case we are talking about class field theory, and I can't recall having seen anything better than Cox's book (already mentioned by Emerton) in this direction (undergrad etc.). With Artin's reciprocity law (the central role of the Frobenius elements in connection with reciprocity laws wasn't exactly kept secret 8-) under your belt you might want to check out


*

*G. Shimura, A reciprocity in law in non-solvable extensions,
J. Reine Angew. Math. 221, 209-220 (1966)


and


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*J. Velu, Lois de reciprocite liees aux courbes elliptiques,
Semin. Delange-Pisot-Poitou, 14e annee 1972/73, Fasc. 1, 2, Expose 9, 5 p. (1973)


both of which are rather nice to read, and then continue with the articles by Serre and Swinnerton-Dyer that Emerton suggested (actually they were motivated by concrete problems concerning Ramanujan's $\tau$-function, and they've written quite a bit about it at the time).
A: If you want to go further in understanding this point of view, I would advise you to begin learning class field theory.  It is a deep subject, it can be understood in a vast variety of ways, from the very concrete and elementary to the very abstract, and although superficially it appears to be limited to describing abelian reciprocity laws, it in fact plays a crucial role in the study of non-abelian reciprocity laws as well.
The texts: 
Ireland and Rosen for basic algebraic number theory, a Galois-theoretic proof
of quadratic reciprocity, and other assorted attractions.
Cox's book on primes of the form x^2 + n y^2 for an indication of what some of the content of class field theory is in elementary terms, via many wonderful examples.
Serre's Local Fields for learning the Galois theory of local fields
Cassels and Frolich for learning global class field theory
The standard book at the graduate level to learn the arithmetic of non-abelian (at least 2-dimensional) reciprocity laws is Modular forms and Fermat's Last Theorem, a textbook on the proof by Taylor and Wiles of FLT.  But it is at a higher level again.
I don't think that you will find a single text on this topic at a basic level (if basic
means Course in arithmetic or Ireland and Rosen), because there is not much to say beyond what you stated in your question without getting into the theory of elliptic curves and/or the theory of modular forms and/or a serious discussion of class field theory.
Also, as basic suggests, you could talk to the grad students in your town, if not at your institution, then at the other one down the Charles river, which as you probably know is currently the world centre for research on non-abelian reciprocity laws (maybe shared with Paris).  Certainly there are grad courses offered on this topic there on a regular basis.
