What is...the Kronecker-Weber theorem?
Goal.
Explaining basic concepts of (a classical course in) algebra in an intuitive way.
This time.
What is...the Kronecker-Weber theorem? Or: Field and Galois theory, application 2
Disclaimer.
Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references.
Slides.
http://www.dtubbenhauer.com/youtube.html
Material used.
https://en.wikipedia.org/wiki/Kroneck...
Root of unity.
https://en.wikipedia.org/wiki/Root_of...
Cyclotomic fields.
https://en.wikipedia.org/wiki/Cycloto...
Kroneckers Jugendtraum.
https://en.wikipedia.org/wiki/Hilbert...
https://mathworld.wolfram.com/Jugendt...
No-go theorem.
https://en.wikipedia.org/wiki/No-go_t...
List of small groups.
https://en.wikipedia.org/wiki/List_of...
Mathematica.
https://mathworld.wolfram.com/RootofU...
https://demonstrations.wolfram.com/Th...
Pictures.
https://mathworld.wolfram.com/RootofU...
To try in Magma.
http://magma.maths.usyd.edu.au/calc/
For some reason youtube doesn't like the angle bracket symbol, so please replace (angle bracket) below by, well, an angle bracket ;-)
P(angle bracket) X (angle bracket):=PolynomialAlgebra(Rationals());
f:=X^4-4*X^2+2;
G:=GaloisGroup(f);
print G;
G=Z/4Z
P(angle bracket) X (angle bracket):=PolynomialAlgebra(Rationals());
f:=X^4-5*X^2+2;
G:=GaloisGroup(f);
print G;
G=D8
#algebra
#galoistheory
#mathematics