Hi, I have a BS in Math, MA in Math, MS in statistics and was a PhD student in math and I must say your videos are the best. All through Algebra 1 and 2; never hear a better explanation on Field Automorphisms and Galois theory, Kudos
I can't get my head around what you mean by switching roots. Surely the polynomial (x - a)(x - b) ... (x - n) is the same no matter how we commute the factors?
Yes, the polynomial itself remains the same when those factors are permuted. (That's actually a feature, not a bug.) What does change is that the number field in which those roots live is transformed onto itself - for instance, if sqrt(2) and -sqrt(2) are interchanged it effects a "reflection" of the field Q(sqrt 2).
How would you suggest I visualise Galois Theory? Linear transformations on the complex plane..? I have only a very basic knowledge of field extensions etc and I'm struggling to find any resources that aren't over my head x) thanks!!
Matthew Salomone, could you please tell me which of your videos is the most direct continuation to this 302.7D lecture? In particular, I would like to understand the connection between abelian-ness of factor groups of a subnormal Galois-group series and the existence of concrete techniques to solve a polynomial with radicals. Would any of the lectures you've posted so far help me to understand this connection specifically? Thanks.
Same (similar) is dual to difference. Symmetry breaking = the Higgs Boson. Bosons like to be in the same state, Fermions like to be in different states:- Bosons are dual to Fermions. Subgroups (quantum, discrete) are dual to subfields (classical, continuous) -- the Galois Correspondence. Galois groups measure how similar (or different) the roots of polynomials are via symmetry breaking (permutations)! "Always two there are" -- Yoda.
Let a,b be 2 (complex) roots of a polynomial p over Q. Is it true that a and b are indistinguishable if and only if the MINIMAL polynomial of a over Q ALSO has b as a root?
+derci ferreira Another excellent question that unfortunately has a long answer, because that's what the rest of the videos in this series attempt to cover. The short version is: try to factor it. If it doesn't factor completely, keep adjoining algebraic irrationals until it does. Then look at how conjugating each of those irrationals affects (or doesn't) the others.
your course is very intersting, i hope to focuse on a new theory such as neutrosophic set pionered by Florentin Smarandache ,he generalized the subgroup...and so on.
Yeah, I'm being purposefully cagey there. Getting more precise about those ideas requires more infrastructure: "similar" is a proxy for "related by an automorphism over the base field." See here, for instance: ua-cam.com/video/HuAbRjnRjmY/v-deo.html
Hi, I have a BS in Math, MA in Math, MS in statistics and was a PhD student in math and I must say your videos are the best. All through Algebra 1 and 2; never hear a better explanation on Field Automorphisms and Galois theory, Kudos
Thank God i found your series of videos. It saved me years of confusion.
I never thought about the idea of trying to "trick" polynomials before; this is all really neat stuff!
Excellent. Your lecture worth a million dollar. Thank you for your excellent contributions to the world of science and mathematics.
An excellent description of how Galois may have thought through this solutions problem. Entertaining and informative.Thanks
Bravo Matt!!
This video is brilliant and it helps me understand why polynomial of order 5 is not solveable (S5 is a single group).
I can't get my head around what you mean by switching roots. Surely the polynomial (x - a)(x - b) ... (x - n) is the same no matter how we commute the factors?
Yes, the polynomial itself remains the same when those factors are permuted. (That's actually a feature, not a bug.) What does change is that the number field in which those roots live is transformed onto itself - for instance, if sqrt(2) and -sqrt(2) are interchanged it effects a "reflection" of the field Q(sqrt 2).
How would you suggest I visualise Galois Theory? Linear transformations on the complex plane..? I have only a very basic knowledge of field extensions etc and I'm struggling to find any resources that aren't over my head x) thanks!!
Try this video from later on in my playlist for ideas. ua-cam.com/video/gbRxT1segUc/v-deo.html
Matthew Salomone, could you please tell me which of your videos is the most direct continuation to this 302.7D lecture? In particular, I would like to understand the connection between abelian-ness of factor groups of a subnormal Galois-group series and the existence of concrete techniques to solve a polynomial with radicals. Would any of the lectures you've posted so far help me to understand this connection specifically? Thanks.
Same (similar) is dual to difference.
Symmetry breaking = the Higgs Boson.
Bosons like to be in the same state, Fermions like to be in different states:- Bosons are dual to Fermions.
Subgroups (quantum, discrete) are dual to subfields (classical, continuous) -- the Galois Correspondence.
Galois groups measure how similar (or different) the roots of polynomials are via symmetry breaking (permutations)!
"Always two there are" -- Yoda.
Let a,b be 2 (complex) roots of a polynomial p over Q. Is it true that a and b are indistinguishable if and only if the MINIMAL polynomial of a over Q ALSO has b as a root?
Great video, really helpful!
how do we know the galois group of a polynomial ,when we dont know its root?
+derci ferreira Another excellent question that unfortunately has a long answer, because that's what the rest of the videos in this series attempt to cover. The short version is: try to factor it. If it doesn't factor completely, keep adjoining algebraic irrationals until it does. Then look at how conjugating each of those irrationals affects (or doesn't) the others.
your course is very intersting, i hope to focuse on a new theory such as neutrosophic set pionered by Florentin Smarandache ,he generalized the subgroup...and so on.
In French it should be: Le reve de Galois (with the accent aigu on the first e of reve...)
What is confusing is that you use the terms "similar roots" and "different roots" without defining these terms.
Yeah, I'm being purposefully cagey there. Getting more precise about those ideas requires more infrastructure: "similar" is a proxy for "related by an automorphism over the base field." See here, for instance: ua-cam.com/video/HuAbRjnRjmY/v-deo.html
Not accent aigu but accent circumflex...
Why people say this video explains Galois theory. Definitely not this video. May be future ones
W