Galois theory: Abel's theorem

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This lecture is part of an online graduate course on Galois theory.

We discuss Abel's theorem, that says a general quintic equation cannot be solved by radicals. We do this by showing that if a polynomial can be solved by radicals over a field of characteristic 0 then its roots lie in a solvable Galois extension. We give some examples of degree 5 polynoimals whose roots do not generate a solvable extension.

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I remenber that Galois proved that a irreducible polynomial of degree p prime is solvable if, and only if, its roots can be expressed in terms of rational polynomials of the others 2 roots, and this gives as a directly corolary the results that you showed here. This also implies that if G is the Galois group of the irreducible polynomial in question, then |G| is less or equal to p(p-1).

If think that this result is kind of ironic, since one of the reasons why Poison discarded the work of Galois was because it lacked a pratical criteria to identify if a given polynomial was solvable.

aa-lrjk

Will there be lectures on Jacobson's theory to draw parallels with Galois one?

maxan

Another question, if a real solution to a polynomial equation of integer coefficients is expressed as "n-th √ of (something involving i)" plus its complex conjugate, is this form considered an expression in terms of radicals?

vs-cwwc

Are roots of 1 always expressible in terms of radicals?

vs-cwwc

Isn't there a little gap in the argument when he aptly points out that ‘n-th roots of all the conjugates need to be adjoined in order to get a normal extension, ’ then pretends adjoining a single one of them gives a galois, thus normal, extension? well, gotta confess, haven't thought about it very carefully.. —great talk otherwise.

nnaammuuss

Subgroups (quantum, discrete) are dual to subfields (classical, continuous) -- the Galois correspondence.
All commutators are cyclic permutations.
"Always two there are" -- Yoda.

hyperduality

so magically you at around 12:09.... turns 5 into zeta and both symbols look the same....is there an explanation why or is that sloppiness on your part?

abdonecbishop

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