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Results tagged with galois-theory
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user 24309
Galois theory, named after Évariste Galois, provides a connection between field theory and group theory. Using Galois theory, certain problems in field theory can be reduced to group theory, which is, in some sense, simpler and better understood.
5
votes
Accepted
Are all rational exactly solvable differential equations known?
I don't think a general answer to your question is known. I personnaly doubt that it could be positive.
Partial decidable answer: if the variational linear differential system obtained along a give …
- 5,428
33
votes
Accepted
What justification can you give for the fact that "most ODEs do not have an explicit solution"?
If the ODE is linear --and the notion of «explicit» refers to Liouvillian solutions (towers of iterated quadrature and exponential of meromorphic functions)-- then its differential Galois group (Picar …
- 5,428
11
votes
The holomorphic version of Galois theory
Say $a_n=1$. You can obtain this map as a section of the map $\sigma$ sending the $n$-tuple of the roots $(r_1,\ldots,r_n)$ to the coefficients of the polynomial using the symmetric polynomials, corre …
- 5,428