Galois deformations are an important tool in Wiles' arsenal for proving FLT. Are there any more elementary aspects (I'm thinking of 1-dimensional Galois representations attached to number fields) that would help the novice in better understanding what's going on?

Here's what I have in mind. Let $\rho: G_{\mathbb Q} \longrightarrow {\mathbb C}^\times$ be a 1-dimensional representation of the absolute Galois group of the rationals factoring over some finite extension. Given a Dirichlet character $\chi: GL_1({\mathbb Z}/N{\mathbb Z}) \longrightarrow {\mathbb C}^\times$, we can find representations $\rho_\chi: Gal(K/{\mathbb Q}) \longrightarrow {\mathbb C}^\times$ for any cyclotomic extension $K = {\mathbb Q}(\zeta_N)$. Call $\rho$ modular if there is a $\chi$ such that $\rho = \rho_\chi$. The statement that every $\rho$ coming from an abelian extension is modular is the theorem of Kronecker-Weber, and in this form it can be proved using Galois deformations along the lines of Wiles' proof (see Tunnell's proof in Kowalski's notes). BTW if anyone knows a source for this result that is more readable than Kowalski's notes (which I discovered just a couple of days ago and haven't studied in detail yet) I'm all ears.

Question: Are there other similarly "elementary" questions, for example in embedding problems or inverse Galois theory, that can be described in terms of Galois deformations?

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