I have heard people say that a major goal of number theory is to understand the absolute Galois group of the rational numbers G = Gal(Q bar/Q). What do people mean when they say this? The Kronecker-Weber theorem gives a good idea of what the abelianization of the G looks like. But in one of Richard Taylor's MSRI talks, Taylor said that he's never heard of anyone proposing a similar direct description of G and that to understand G one studies the representations of G.

I know that there is a strong interest in showing the Langlands reciprocity conjecture [Edit: What I had in mind in writing this is evidently Clozel's conjecture, not the Langlands reciprocity conjecture - see Kevin Buzzard's post below] - that L-functions attached to l-adic Galois representations coincide with L-functions attached to certain automorphic representations. And I've heard people refer to the Tannakian philosophy which I understand as (roughly speaking) asserting that G is determined by all of its finite dimensional representations.

Here is a representation of G understood not to be a representation of G as an abstract group but as a group together with a labeling of some of the conjugacy classes of G by rational primes (the Frobenius elements)?

When people talk about "understanding G" do they mean proving [Edit: Clozel's conjecture] (in view of the Tannakian philosophy)? If not, what do they mean? If so, this conceptualization seems quite abstract to me. Is this what people mean when they say "understand G"? Can [Edit: Clozel's conjecture] be used to give more tangible statements about G?

Something that I have in mind as I write this is the inverse Galois problem (does every finite group occur as a Galois group of a normal extension of Q?) and Gross' conjecture (mostly proven by now) that for each prime p there exists a nonsolvable extension of Q ramified only at p. But I am open to and interested in other senses and respects in which one might "understand" G

I have heard people say that a major goal of number theory is to understand the absolute Galois group of the rational numbers $G = \mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$. What do people mean when they say this? The Kronecker-Weber theorem gives a good idea of what the abelianization of $G$ looks like. But in one of Richard Taylor's MSRI talks, Taylor said that he's never heard of anyone proposing a similar direct description of $G$ and that to understand $G$ one studies the representations of $G$.

I know that there is a strong interest in showing the Langlands reciprocity conjecture [Edit: What I had in mind in writing this is evidently Clozel's conjecture, not the Langlands reciprocity conjecture - see Kevin Buzzard's post below] - that $L$-functions attached to $\ell$-adic Galois representations coincide with $L$-functions attached to certain automorphic representations. And I've heard people refer to the Tannakian philosophy which I understand as (roughly speaking) asserting that $G$ is determined by all of its finite dimensional representations.

Here is a representation of $G$ understood not to be a representation of $G$ as an abstract group but as a group together with a labeling of some of the conjugacy classes of G by rational primes (the Frobenius elements)?

When people talk about "understanding $G$" do they mean proving [Edit: Clozel's conjecture] (in view of the Tannakian philosophy)? If not, what do they mean? If so, this conceptualization seems quite abstract to me. Is this what people mean when they say "understand $G$"? Can [Edit: Clozel's conjecture] be used to give more tangible statements about $G$?

Something that I have in mind as I write this is the inverse Galois problem (does every finite group occur as a Galois group of a normal extension of $\mathbb{Q}$?) and Gross' conjecture (mostly proven by now) that for each prime $p$ there exists a nonsolvable extension of $\mathbb{Q}$ ramified only at $p$. But I am open to and interested in other senses and respects in which one might "understand" $G$.

I have heard people say that a major goal of number theory is to understand the absolute Galois group of the rational numbers $G = Gal(Q bar/Q)$. What do people mean when they say this? The Kronecker-Weber theorem gives a good idea of what the abelianization of the $G$ looks like. But in one of Richard Taylor's MSRI talks, Taylor said that he's never heard of anyone proposing a similar direct description of $G$ and that to understand G one studies the representations of $G$.

I know that there is a strong interest in showing the Langlands reciprocity conjecture [Edit: What I had in mind in writing this is evidently Clozel's conjecture, not the Langlands reciprocity conjecture - see Kevin Buzzard's post below] - that $L$-functions attached to $l$-adic Galois representations coincide with $L$-functions attached to certain automorphic representations. And I've heard people refer to the Tannakian philosophy which I understand as (roughly speaking) asserting that $G$ is determined by all of its finite dimensional representations.

Here is a representation of G understood not to be a representation of $G$ as an abstract group but as a group together with a labeling of some of the conjugacy classes of $G$ by rational primes (the Frobenius elements)?

When people talk about "understanding $G$" do they mean proving [Edit: Clozel's conjecture] (in view of the Tannakian philosophy)? If not, what do they mean? If so, this conceptualization seems quite abstract to me. Is this what people mean when they say "understand $G$"? Can [Edit: Clozel's conjecture] be used to give more tangible statements about $G$?

Something that I have in mind as I write this is the inverse Galois problem (does every finite group occur as a Galois group of a normal extension of $Q$?) and Gross' conjecture (mostly proven by now) that for each prime $p$ there exists a nonsolvable extension of $Q$ ramified only at $p$. But I am open to and interested in other senses and respects in which one might "understand" G

I have heard people say that a major goal of number theory is to understand the absolute Galois group of the rational numbers G = Gal(Q bar/Q). What do people mean when they say this? The Kronecker-Weber theorem gives a good idea of what the abelianization of the G looks like. But in one of Richard Taylor's MSRI talks, Taylor said that he's never heard of anyone proposing a similar direct description of G and that to understand G one studies the representations of G.

I know that there is a strong interest in showing the Langlands reciprocity conjecture [Edit: What I had in mind in writing this is evidently Clozel's conjecture, not the Langlands reciprocity conjecture - see Kevin Buzzard's post below] - that L-functions attached to l-adic Galois representations coincide with L-functions attached to certain automorphic representations. And I've heard people refer to the Tannakian philosophy which I understand as (roughly speaking) asserting that G is determined by all of its finite dimensional representations.

Here is a representation of G understood not to be a representation of G as an abstract group but as a group together with a labeling of some of the conjugacy classes of G by rational primes (the Frobenius elements)?

When people talk about "understanding G" do they mean proving [Edit: Clozel's conjecture] (in view of the Tannakian philosophy)? If not, what do they mean? If so, this conceptualization seems quite abstract to me. Is this what people mean when they say "understand G"? Can [Edit: Clozel's conjecture] be used to give more tangible statements about G?

Something that I have in mind as I write this is the inverse Galois problem (does every finite group occur as a Galois group of a normal extension of Q?) and Gross' conjecture (mostly proven by now) that for each prime p there exists a nonsolvable extension of Q ramified only at p. But I am open to and interested in other senses and respects in which one might "understand" G