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Peter Humphries
Peter Humphries
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Taylor said that he's never heard of anyone proposing a similar direct description of G$G$ and that to understand G$G$ one studies the representations of G$G$.

I remember Mazur telling me this when I was a grad student. HeHe made this point in the following way. YouYou shouldn't really think of Gal(Qbar/Q)$\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ as a group which has elements, but as a "group up to conjugacy" -- thus, the aspects of Galois groups that really make sense to think about are the conjugacy-invariant things: conjugacyconjugacy classes (like Frobenii) and representations.

To unpack this a bit more: aa Galois group is a fundamental group. ButBut to talk about a fundamental group (as opposed to a groupoid) you need to choose a basepoint. ToTo talk about an absolute Galois group you also need to choose a basepoint, which is to say an algebraic closure Qbar/Q$\bar{\mathbb{Q}}/\mathbb{Q}$.  (So just as one should talk about pi_1(X,*)$\pi_1(X,*)$ rather than pi_1(X)$\pi_1(X)$, one should talk about Gal(Qbar/Q)$\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ rather than Gal(Q)$\mathrm{Gal}(\mathbb{Q})$.) ButBut a basepoint you can just draw with a pencil. AA Galois closure of Q$\mathbb{Q}$ is not so easy.

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 remember Mazur telling me this when I was a grad student. He made this point in the following way. You shouldn't really think of Gal(Qbar/Q) as a group which has elements, but as a "group up to conjugacy" -- thus, the aspects of Galois groups that really make sense to think about are the conjugacy-invariant things: conjugacy classes (like Frobenii) and representations.

To unpack this a bit more: a Galois group is a fundamental group. But to talk about a fundamental group (as opposed to a groupoid) you need to choose a basepoint. To talk about an absolute Galois group you also need to choose a basepoint, which is to say an algebraic closure Qbar/Q.  (So just as one should talk about pi_1(X,*) rather than pi_1(X), one should talk about Gal(Qbar/Q) rather than Gal(Q).) But a basepoint you can just draw with a pencil. A Galois closure of Q is not so easy.

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 remember Mazur telling me this when I was a grad student. He made this point in the following way. You shouldn't really think of $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ as a group which has elements, but as a "group up to conjugacy" - thus, the aspects of Galois groups that really make sense to think about are the conjugacy-invariant things: conjugacy classes (like Frobenii) and representations.

To unpack this a bit more: a Galois group is a fundamental group. But to talk about a fundamental group (as opposed to a groupoid) you need to choose a basepoint. To talk about an absolute Galois group you also need to choose a basepoint, which is to say an algebraic closure $\bar{\mathbb{Q}}/\mathbb{Q}$. (So just as one should talk about $\pi_1(X,*)$ rather than $\pi_1(X)$, one should talk about $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ rather than $\mathrm{Gal}(\mathbb{Q})$.) But a basepoint you can just draw with a pencil. A Galois closure of $\mathbb{Q}$ is not so easy.

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JSE
JSE
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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 remember Mazur telling me this when I was a grad student. He made this point in the following way. You shouldn't really think of Gal(Qbar/Q) as a group which has elements, but as a "group up to conjugacy" -- thus, the aspects of Galois groups that really make sense to think about are the conjugacy-invariant things: conjugacy classes (like Frobenii) and representations.

To unpack this a bit more: a Galois group is a fundamental group. But to talk about a fundamental group (as opposed to a groupoid) you need to choose a basepoint. To talk about an absolute Galois group you also need to choose a basepoint, which is to say an algebraic closure Qbar/Q. (So just as one should talk about pi_1(X,*) rather than pi_1(X), one should talk about Gal(Qbar/Q) rather than Gal(Q).) But a basepoint you can just draw with a pencil. A Galois closure of Q is not so easy.