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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.
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
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