Galois Theory for Beginners (2010) [pdf]
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Galois Theory for Beginners (2010) [pdf]
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Re: Galois Theory for Beginners (2010) [pdf]
#2Fun fact: Evariste Galois made major contributions to math in his teens, before dying in a duel at age 20.
Re: Galois Theory for Beginners (2010) [pdf]
#3Fun 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
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.
Re: Galois Theory for Beginners (2010) [pdf]
#4Fun 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
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.
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
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.
Re: Galois Theory for Beginners (2010) [pdf]
#5this guy was AMAZING at maths
Re: Galois Theory for Beginners (2010) [pdf]
#6Galois 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
PS: He has an entire series of lectures on his channel. Highly recommend.
Re: Galois Theory for Beginners (2010) [pdf]
#7Fun 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
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.
Tony Rothman wrote an article in the American Mathematical Monthly which helped to separate a lot of the fact from the fiction in accounts of Galois's life. Highly recommended:
https://www.maa.org/sites/default/files/pdf/upload_library/2...
Re: Galois Theory for Beginners (2010) [pdf]
#8Can 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).
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
sigma f(5, 3) = sigma (5 - 3) = sigma 2 = f(3, 5) = -2
sigma g(5, 3) = sigma 2 = g(3, 5) = 2
Re: Galois Theory for Beginners (2010) [pdf]
#9Can 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 Th…
Your sigma is a bijection on rational functions of a, b, not on the evaluations of those functions at particular values of a and b. In lambda notation:*
sigma f = \a b -> f (sigma a) (sigma b) = f b a
That means sigma f = \a b -> b - a. On the other hand, g := \a b -> 2. So sigma g = g.
Looking again at your equality:
> sigma f(5, 3) = sigma (5 - 3) = sigma 2 = f(3, 5) = -2
We should read sigma f(5, 3) as (sigma f)(5,3) = (3 - 5) = -2. Note: sigma f is not equal to sigma (5 - 3), because f is not the same rational function as "5 - 3"!
* This is an abuse of notation, because a and b are bound variables inside the lambda. In this case, being more precise would probably be less clear.
Re: Galois Theory for Beginners (2010) [pdf]
#10Can 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 Th…
I am a little confused by your example, but the definition
σf(x_1,...,x_n) = f(σx_1,...,σx_n)
I'm really not sure what σ(2) is supposed to represent? σ acts on rational functions like f and g, so
σf(5, 3) = f(3, 5) = -2
σg(5,3) = g(3, 5) = 2