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Good morning, today I have read that "number theory is nothing but the study of $\mathrm{Gal}(\mathbb{\bar{Q}}/\mathbb{Q})$", here http://www.math.uconn.edu/~alozano/elliptic/finding%20points.pdf can anyone give a very naive layman definition of what it actually means?

Furthermore, I got this doubt that $\bar {\mathbb{Q}}$ is the algebraic closure of $\mathbb{Q}$, and the thing that confuses me is the field of rational numbers $\mathbb{Q}$ is not a algebraically closed as there exists a polynomial with $a_{1},a_{2},\dotsc,a_{n}\in \mathbb{Q}$ and $(x-a_{1})(x-a_{2})\cdots(x-a_{n})+1$ has no zero in $\mathbb{Q}$.

Then why are we considering the field extension of $\bar {\mathbb{Q}}/\mathbb{Q}$ when $\mathbb{Q}$ is not algebraically closed, won't it contradict the definition of algebraic closure?

But I am not getting an answer i was looking for ,i want what are the things going on behind the $\mathrm{Gal}(\mathbb{\bar{Q}}/\mathbb{Q})$ like what is the thing we get if we take the $\mathbb{\bar{Q}/\mathbb{Q}}$ and what does taking the $\mathrm{Gal}(\mathbb{\bar{Q}}/\mathbb{Q})$ give someone,

Thank you

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$\overline{\mathbb{Q}}$ is algebraically closed, $\mathbb{Q}$ isn't, and it would be very boring Galois theory if the base field were closed. –  Zhen Lin Apr 27 '11 at 8:56
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Why did this question get a downvote? –  Stijn Apr 27 '11 at 8:57
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@Stijn When someone clearly doesn't understand what algebraic closure is, but asks some questions about Gal(\bar Q/Q)... –  Grigory M Apr 28 '11 at 17:07
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@Grigory M: mr grigory i felt very bad looking at your comment,this is a place where one shares and gains the knowledge,i dont know the galois group,and extensions so thats why i am asking it here,if i would have known these things why would i ask ?, and you must accept that no one is know all in the universe everyone die as a student as there are lot many things to be learnt,and there was a time when you are a child you even dont know numbers,and alphabets,now you know them,because some one had told you and made you learn them –  Iyengar Apr 29 '11 at 3:59
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Perhaps the reason you're not getting an answer you're looking for is you're not asking the question you mean to ask. No one can understand what you mean by "what is the thing we get if we take the Q-bar over Q?" Now if what you really want to do is ask, "what does Q-bar mean?" and/or "what does Gal mean?" then you can ask those questions and someone will be happy to answer you. But we're just guessing what you want to know because it's hard for us to make sense of your questions. –  Gerry Myerson Apr 29 '11 at 11:42
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4 Answers

up vote 20 down vote accepted

Perhaps the most accessible introduction for a "very naive layperson" is Ash and Gross's Fearless Symmetry. I recall reading many glowing reviews from non-experts, so this may be precisely the exposition that you seek. See also Gross's links to 10 reviews. I think you will find some of these reviews quite informative.

You may also find of interest this excerpt from an interview with Richard Taylor.

WHAT ARE YOUR MAJOR RESEARCH INTERESTS AND ACHIEVEMENTS?

The great problem that motivates me is to understand the absolute Galois group of the rational numbers, that is, the group of all automorphisms of the field of algebraic numbers (complex numbers which are the roots of nonzero polynomials with rational coefficients). If you like you can talk about all Galois groups of finite extensions of the rational numbers, but this is a convenient way to put them all together. It doesn't make a lot of difference, but it is technically neater to put them all together. The question that has motivated almost everything I have done is, "What's the structure of that group?" One of the great achievements of mathematicians of the first half of this century is called class field theory, and one way of seeing it is as a description of all abelian quotients of the absolute Galois group of Q, or if you like, the classification of the abelian extensions of the field of the rational numbers. That's only a very small part of this group. The group is extremely complicated, and just describing the abelian part doesn't solve the problem. For instance John Thompson proved that the monster group is a quotient group of this group in infinitely many ways.

There is some sort of program to understand the rest of this group, often referred to as the Langlands Program. There's a huge mass of conjectures, of which we are only beginning to scratch the surface, which tell us what the structure is. The answer is to my mind extremely surprising; it invokes extremely different objects. You start out with this algebraic structure and end up using what are called modular forms, which relate to complex analysis.

There seems to be an answer to this question: what's the structure? And the answer is something completely unexpected in terms of these analytic objects, and I think that's what attracts me to the subject. When there is a great connection between two different areas of mathematics, it always seems to me indicative that something interesting is going on.

The other thing we can see--another indication that it's a powerful theory--is that one can answer questions one might have asked anyway, before one built up the theory. Maybe, the first example was a result proved by Barry Mazur; he provided a description of the possible torsion subgroups of elliptic curves defined over the rational numbers. It was a problem that had been knocking around for some time, and it's relatively easy to state. Using these sorts of ideas, Barry was able to settle it.

Other examples are the proof the main conjecture of Iwasawa theory by Barry Mazur and Andrew Wiles, and the work of Dick Gross and Don Zagier on rational points on elliptic curves. And I guess finally, there's Fermat's last theorem, which Andrew Wiles solved using these ideas again. So in fact, the story of Fermat's last theorem is that this German mathematician Frey realized that if you knew enough of this correspondence between modular forms and Galois groups, there is an extraordinarily quick proof of Fermat's last theorem. And at the time he realized this, not enough was known about this correspondence. What Andrew Wiles did and Andrew and I completed was prove enough about this correspondence for Frey's argument to go through. The thing that amuses me is that it seems that history could easily have been reversed. All these things could have been proved about the relationship between modular forms and Galois groups, and then Frey could have come along and given nearly a two-line proof of Fermat's last theorem.

Those four [torsion points, Iwasawa theory, Gross and Zagier, Fermat] are probably the obvious big applications of these sorts of ideas. It seems to me the applications have been extraordinarily successful--at least four things that would have been recognized as important problems irrespective of this theory, problems that people had thought about before modular forms.

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it was quite useful –  Iyengar Apr 29 '11 at 4:03
    
Well said Prof.Taylor :) –  Dinesh May 7 '11 at 20:37
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You might want to let us know where you read that quote, the context could be helpful in determining the author's intentions. I'm guessing that it wasn't meant to be taken too seriously, and I'd recommend you take it to mean only that there are a few things in Number Theory that you can understand better if you know something about that Galois group.

If you want to look into it further, a good keyphrase is "absolute Galois group."

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If you look at the OP's profile, you will see that he is not particularly fond of mathematicians' "jargon". So by giving him that keyphrase, you have just revealed yourself as another one of those mathematicians who try to make their work look complicated. –  Alex B. Apr 27 '11 at 13:14
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@Alex, I can live with that. –  Gerry Myerson Apr 27 '11 at 13:22
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perhaps it was here. –  Zev Chonoles Apr 27 '11 at 13:59
    
@Zev thanks! @Gerry Interesting comment :-) Why dont you please edit your answer by giving a simple example or a situation where the connection can be seen. Perhaps a naive one would suffice. thanks. –  Dinesh May 7 '11 at 20:41
    
@Dinesh, sorry, I don't even know a naive example of something in Number Theory that gets explained well by the absolute Galois group. I repeat my suggestion to type that phrase into your favorite search engine and see what turns up. –  Gerry Myerson May 8 '11 at 1:48
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I would say it's not unreasonable to (loosely) describe algebraic number theory as the study of number fields (number fields are the objects you get when you adjoin roots of polynomials like $X^2+1$ to $\mathbb{Q}$). This is admittedly an incredibly vague description and could be taken in any number of ways, but, if you allow it, then, via Galois theory (for infinite degree extensions of $\mathbb{Q}$, like $\bar{\mathbb{Q}}$), Galois number fields are (roughly) in bijection with finite quotients of the Galois group of $\bar{\mathbb{Q}}$ over $\mathbb{Q}$. (I say roughly because really the bijection is with normal subgroups of finite index of the Galois group that are closed for a certain topology.) For instance, a sort of natural question in algebraic number theory would be "what finite groups occur as Galois groups of Galois field extensions of $\mathbb{Q}$?" This is equivalent to the question "what are the finite quotients of the Galois group of $\bar{\mathbb{Q}}$ over $\mathbb{Q}$?"

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The object $\mathrm{Gal}(\mathbb{\bar{Q}}/\mathbb{Q})$ is not just an abstract group; it has a natural and intricate topology on it as well. Then one would consider various naturally occurring continuous actions of this topological group on various modules. Two examples are: The absolute Galois group of a number field acts naturally on the roots of unity contained in that field, or, given an elliptic curve over the field, the Galois group acts on the torsion points on the elliptic curve with co-ordinates contained in the field.

Also one considers some natural representations of this Galois group on some vector spaces. Examples can be constructed from the above description of Galois actions. A lot of number theory is encapsulated in the study of such Galois actions and Galois representations. This is perhaps what is meant by the statement that you quote.

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