
Galois Theory for Beginners (2010) [pdf] - fanf2
http://www.math.jhu.edu/~smahanta/Teaching/Spring10/Stillwell.pdf
======
ktta
Galois fields are used in crypto a bit, and I recommend this lecture to
understand it in the context of crypto

[https://www.youtube.com/watch?v=x1v2tX4_dkQ](https://www.youtube.com/watch?v=x1v2tX4_dkQ)

PS: He has an entire series of lectures on his channel. Highly recommend.

~~~
drevet
Thanks ;)

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kdamica
Fun fact: Evariste Galois made major contributions to math in his teens,
before dying in a duel at age 20.

[https://en.wikipedia.org/wiki/%C3%89variste_Galois](https://en.wikipedia.org/wiki/%C3%89variste_Galois)

~~~
schoen
Wikipedia reminds me that the version of his death (staying up all night
before the duel to frantically write down his mathematical ideas at the last
possible moment) that I read in _Men of Mathematics_ may have been
overdramatized, apparently like a number of the stories in that book.

~~~
DonbunEf7
What definitely did happen is that Galois was worried that he might not win
the duel and that he might never get the chance to redeem himself in front of
his mathematical peers. We know this because of the letter he wrote before he
died; we actually _have_ it, and WP has a scan of a page:
[https://en.wikipedia.org/wiki/File:E._Galois_Letter.jpg](https://en.wikipedia.org/wiki/File:E._Galois_Letter.jpg)

It is hard to overdramatize the events of Galois' life. He lived during a
revolution, he went out into the streets to protest and fight, he was arrested
multiple times for his outrageous political speech, and he laid down his life
for his political beliefs. A friend of mine who survived a chronic and often-
fatal disease in his teens used to quip, "Well, if I were Galois, I'd have
already made my best contributions and died by now." Galois indeed burned fast
and bright.

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netvarun
This document was also mentioned yesterday on the 'Why is the quintic
unsolvable?'[1] post. The post also has some additional interesting references
in the comments. Worth checking out!

[1]
[https://news.ycombinator.com/item?id=14685466](https://news.ycombinator.com/item?id=14685466)

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monfrere
Can anyone help me understand what is happening at the bottom of page 23 (page
3 of the PDF)?

It says any permutation sigma of x_1, ..., x_n can be extended to a bijection
of Q(x_1, ..., x_n) defined by

    
    
        sigma f(x_1, ..., x_n) = f(sigma x_1, ..., sigma x_n).
    

But I don't see how this definition can be consistent. For example, let

    
    
        f(a, b) = a - b
        g(a, b) = a/a + b/b = 2
    
        x_1 = 5
        x_2 = 3
        sigma x_1 = x_2
        sigma x_2 = x_1
    

Then according to the formula:

    
    
        sigma f(5, 3) = sigma (5 - 3) = sigma 2 = f(3, 5) = -2
    

But

    
    
        sigma g(5, 3) = sigma 2 = g(3, 5) = 2
    

Contradiction?

~~~
ky3
Great observation.

The American Math Monthly is a journal by college professors for college
professors. The readership is expected to be familiar with the topic.
Groundbreaking results are published elsewhere. Whew. Unfortunately, neither
is AMM a place for professors to summarize the contents of a 14-week course
for adult learners.

Let's use your observation to illuminate 2 abuses of notation that happen all
the time between those in the know.

The first abuse is not explicitly calling out that the coefficients a_i are
restricted. The polynomial of which the a_i are the coefficients must be
irreducible. That is omitted in the paper, but is typically understood. For
otherwise, the field extension doesn't work, as you've found out.

When the a_i denote an irreducible, then the roots x_i are all outside Q.

And then there is no contradiction.

Your example uses (x-5)(x-3) which has all roots in Q--the diametric opposite
--which is why sigma breaks down.

Digression: If you know some Haskell, you'll notice that a permutation on the
roots basically fmaps to a (field) endomorphism on Q(all x_i). But here the
converse is also true (exceptional in Haskell, except for trivial cases):
every such endo comes from a permutation. (end of digression)

The second abuse is in the title. This is really "(My Opinion on) How to Teach
Galois Theory to Undergrads" with a subtitle of "By Jettisoning the
Fundamental Theorem and Focusing Exclusively on Quintic Unsolvability." The
subtitle is omitted and the title shortened and de-colloquialized to read
"Galois Theory for Beginners." This is all part of the prestigious
mathematical tradition because ink, paper, and papyrus once upon a time were
terribly scarce. Sorry about that.

Quintic Unsolvability is like FLT. The big prize is not the Yes/No answer but
the VIP theorems--the statements of which are neither as easy to explain nor
understand as QU nor FLT--used to nail down some pesky boolean.

So throwing out the FT of GT shortchanges the undergrad. It especially
shortchanges the math-aware software professional who would appreciate
experiencing the galois correspondence which later morphs into an adjunction
in category theory. Quite cool.

GT has pedagogical messiness like inseparable extensions which can be skipped
on a first pass. As a royal road to FTGT, I recommend the approach of fixing
all fields as subfields of the complex numbers. See Postnikov's Foundations of
Galois Theory available on google books the last time I checked. Nice
exercises too.

p.s. (Galois) adjunctions are like a general theory of "How to Run Anything
Backwards Even When There's No Chance in Hell." That's the power of math for
you.

~~~
monfrere
OK, thanks. Your answer has allowed me to follow the paper a little further,
though I think your answer may be inconsistent with the other two answers I
got. Also I haven't been able to show myself that if the x_i are the roots of
an irreducible polynomial, then Q(x_1,...,x_n) is symmetric with respect to
x_1,...,x_n, in the sense claimed in the paper without proof.

~~~
ky3
> OK, thanks.

No worries.

> Your answer has allowed me to follow the paper a little further, though I
> think your answer may be inconsistent with the other two answers I got.

I was trying to be helpful.

That said, in lieu of the irreducibility criterion, you could stay in the
original context where all the variables a_i, x_i, and alpha_i are
indeterminates so that all the extensions are higher-ranked rational function
fields over Q.

So Q(a1,a2) is a field in 2 indeterminates and Q(x1,x2) is an extension of
that.

They are isomorphic to subfields of C, but we don't think of them as subfields
of C because there don't come with canonical embeddings. You have to choose
the algebraically independent transcendentals.

> Also I haven't been able to show myself that if the x_i are the roots of an
> irreducible polynomial, then Q(x_1,...,x_n) is symmetric with respect to
> x_1,...,x_n, in the sense claimed in the paper without proof.

There's no claim. The paper's just defining what it means for a field
extension to be "symmetric w.r.t." to the adjoined elements.

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gerbilly
Galois had an insight that I always seemed particularly deep to me: that
problems should be classified not by topic area (analysis, theory of
equations, geometry) but by their underlying form.

Galois was ahead of his time.[1]

[1] You could say he was ahead by a century, to quote a famous song.

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empath75
Beginners?

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Ceezy
Galois theory without functor... Without fundamental theorem of algebra. Are
you kidding?

~~~
stablemap
It's a five page article in the _Monthly_ , but I'd hesitate to include either
topic even in a book.

~~~
Ceezy
Why? Whitout a clear explanation Galois functors you miss everything... And
without the fundamental theorem of algebra, it's going to be very hard to have
any real life example.

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ManyEthers
I have never studied group theory, so this is way beyond what I'm ready for.
But I scanned over a few sections and read to the point that I got lost.

The part that surprised me is that it seems to focus on rational numbers. I
always assumed that Galois Groups were focused on more abstract concepts of
sets. Is Galois Theory mostly about rational number (or even real numbers), or
is the author just using the rationals to keep the paper focused on beginners?

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mdevere
this guy was AMAZING at maths

