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Galois theory - introductory text from geometric viewpoint?
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John Baez
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Joined: 01 May 2005
Posts: 220

PostPosted: Tue Jul 18, 2006 12:45 pm    Post subject: Galois theory - introductory text from geometric viewpoint? Reply with quote

Does anyone know an introductory text on Galois theory that
really emphasizes the analogy with the theory of branched
covering spaces? This book makes the analogy very clear:

Dino Lorenzini, An Invitation to Arithmetic Geometry, American
Mathematical Society, Providence, Rhode Island, 1996.

But, it's not what I'd call introductory. Michio Kuga's
book Galois' Dream seems to emphasize covering spaces, but
it's more about Fuchsian differential equations than the
Galois theory of number fields, which is what I'm talking
about here.

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laurent berger
science forum beginner


Joined: 23 May 2006
Posts: 3

PostPosted: Tue Jul 18, 2006 1:45 pm    Post subject: Re: Galois theory - introductory text from geometric viewpoint? Reply with quote

John Baez a Žcrit :

Quote:
Does anyone know an introductory text on Galois theory that
really emphasizes the analogy with the theory of branched
covering spaces?

There is the following book :

"Algbre et thŽories galoisiennes"

By Adrien Douady and RŽgine Douady - Cassini; Avril 2005

it's a reissue of their 1978 (?) book.

Best regards

Laurent Berger
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Maarten Bergvelt
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Joined: 23 May 2005
Posts: 22

PostPosted: Tue Jul 18, 2006 2:59 pm    Post subject: Re: Galois theory - introductory text from geometric viewpoint? Reply with quote

In article <e9iokt$tp$1@dizzy.math.ohio-state.edu>, laurent berger wrote:
Quote:


John Baez a åÚcrit :

Does anyone know an introductory text on Galois theory that
really emphasizes the analogy with the theory of branched
covering spaces?

There is the following book :

"Algå¶bre et thåÚories galoisiennes"

By Adrien Douady and RåÚgine Douady - Cassini; Avril 2005

it's a reissue of their 1978 (?) book.

I liked these books. From Math Reviews MR0595327 (82b:12024a):
----
These two volumes provide a rich, dense, unrelentingly Bourbakian
approach to both algebraic Galois theory (finite and infinite) and the
analogous topological theory (coverings of Riemann surfaces).
......
Finally it must be said that while these volumes are ostensibly
intended as a textbook ("Ce livre s'adresse aux tudiants de 2e
ann e de matrise"), only the most extraordinarily able and highly
motivated student, steeped in rigorous abstraction from infancy, could
be expected to really learn the material covered in these books, ab
initio, from them. On the other hand they are valuable as a concise,
yet thorough, presentation of Galois theory, algebraic and
topological, in quite an abstract context, and they will serve as a
useful resource for readers possessing sufficient sophistication to
appreciate them.
-----
Maarten Bergvelt
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Timothy Murphy
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Joined: 29 Apr 2005
Posts: 275

PostPosted: Tue Jul 18, 2006 7:18 pm    Post subject: Re: Galois theory - introductory text from geometric viewpoint? Reply with quote

John Baez wrote:

Quote:
Michio Kuga's
book Galois' Dream seems to emphasize covering spaces, but
it's more about Fuchsian differential equations than the
Galois theory of number fields, which is what I'm talking
about here.

Is Grothendieck's Dessins d'Enfants relevant?

--
Timothy Murphy
e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie
tel: +353-86-2336090, +353-1-2842366
s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland
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John Baez
science forum Guru Wannabe


Joined: 01 May 2005
Posts: 220

PostPosted: Wed Jul 19, 2006 12:02 pm    Post subject: Re: Galois theory - introductory text from geometric viewpoint? Reply with quote

In article <e9isvu$1d2$1@dizzy.math.ohio-state.edu>,
Maarten Bergvelt <bergv@uiuc.edu> wrote:

Quote:
In article <e9iokt$tp$1@dizzy.math.ohio-state.edu>, laurent berger wrote:

John Baez a åÚcrit :

Does anyone know an introductory text on Galois theory that
really emphasizes the analogy with the theory of branched
covering spaces?

There is the following book :

"Algebre et theories galoisiennes"

By Adrien Douady and RåÚgine Douady - Cassini; Avril 2005

That sounds interesting! But can anyone recommend a textbook on
Galois theory that treats the analogy to branched covering spaces
and is not "dense and unrelentingly Bourbakian" - suitable for kids
OTHER THAN "the most extraordinarily able and highly motivated
student, steeped in rigorous abstraction from infancy"???

The analogy to geometry is supposed to make Galois theory EASIER to
understand. You can draw pictures and stuff.

Since I was steeped in rigorous abstraction from infancy, I enjoy
dense and unrelentingly Bourbakian texts for bedtime reading.
But right now I'm looking for a book that someone ELSE might enjoy -
someone who doesn't already know Galois theory.
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