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- Galois theory on graphs - Mathematics Stack Exchange
There is actually an exact analogue of Galois theory in this context, given by the theory of covering spaces in topology Covering space theory defines a topological version of a (separable) field extension called a covering space, and specialized to graphs, the covering spaces of a graph are always graphs and can be defined purely combinatorially - these are covering graphs One can give a
- galois theory - How to solve polynomials? - Mathematics Stack Exchange
14 Using Galois theory we can effectively compute whether or not a polynomial is solvable in radicals - technically this means you can build a chain of field extensions by adding n n -th roots of previously defined elements Anyway I was wondering, how do we actually solve the polynomials when they can be solved?
- Example of Galois extension whose Galois group is $A_4$
With very high probability (actually, essentially 100%, in a sense that can be made precise), its splitting field K K has Galois group S4 S 4 over Q Q Now since Δ−−√ Δ is the product of the differences of the roots of f f, it lives in K K, and so whenever Δ Δ is not a perfect square, Q(Δ−−√) Q (Δ) is a quadratic extension of
- A good way to understand Galois covering? - Mathematics Stack Exchange
A covering map f: X → Y f: X → Y is called Galois if for each y ∈ Y y ∈ Y and each pair of lifts x,x x, x, there is a covering transformation taking x x to x x What is a good way to understand this definition? It seems to me that f f is Galois if and only if Y Y is obtained from X X as a quotient of some group
- Original works of great mathematician Évariste Galois
Galois explained that "rational" in his work would mean a quantity expressible in terms of (ordinary) rational numbers, the coefficients of a given equation, and "any other quantities that we have adjoined (to the equation) " Galois introduced the word "group" to refer to groups of permutations of roots of an equation
- Where can I find Galois original paper? - Mathematics Stack Exchange
As we all know Galois is an ultimate math prodigy At age 17 or 18 he published a paper which we now know as Galois theory I want to just see how he thought mathematics by seeing his original pape
- Galois group of $x^3-x-1$ over $\\mathbb{Q}(i\\sqrt{23})$
I know the Galois group of x3 − x − 1 x 3 x 1 over Q Q is S3 S 3 But to find the Galois group over Q(i 23−−√) Q (i 23) we need to find a splitting field
- Finding Galois extension whose Galois group is Sn
I want to prove that given any integer n, we can find a finite Galois extension K over Q Q such Gal (K: Q) (K: Q) = Sn S n For prime p, I know finding a polynomial with exactly 2 nonreal roots will have Galois group of splitting field Sp S p Can we find infinitely many such polynomials for prime p? What about composite n?
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