Questions tagged [galois-theory]
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.
887 questions
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Fallback for failure case in Galois factoring with units
Let $N = x^2 + 3y^2$ be a composite integer with a known representation of this form. Consider the cubic polynomial
$$
f(t) = 4t^3 - 3Nt - Nx,
$$
and let $K = \mathbb{Q}(\alpha)$ be the cubic number ...
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Is the absolute Galois group of $k(t)$ a semi direct product?
I asked the following question already on Math-Stackexchange here but it might be better fitted here.
Let $k$ be a field (assume characteristic $0$ to make life a bit simpler) and let $K = k(t)$. Then ...
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Quartic to octic extensions for fields
If a field has a quartic extension, must it also have an octic extension?
I believe the answer is no, and even have a possible example field $F$ in mind. Fix an algebraic closure $\overline{\mathbb{Q}...
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If a finite Galois extension of a number field with given Galois group exists, can it always be chosen to be étale at a finite set of primes?
The general question whether or not for a given finite group $G$ and a given number field $K$ there exists a Galois extension $L/K$ with $\operatorname{Gal}(L/K)\cong G$ is as of now unresolved.
But ...
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If $r_1,\dots,r_n$ are Galois-conjugate, can $\sum_i\sigma(r_i)/r_i$ be constant?
Let $f(x)\in\mathbb{Z}[x]$ be a monic irreducible polynomial of degree $n\geq2$ with roots $r_1,\dots,r_n$. Let $K=\mathbb{Q}(r_1,\dots,r_n)$ be its splitting field over $\mathbb{Q}$ with Galois group ...
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Factorization of a parameter-dependent quintic polynomial arising from a photon-sphere equation
I am studying a parameter–dependent quintic polynomial that arises from a dimensionless “master equation” for the photon sphere in a certain black hole model. After nondimensionalization and ...
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Approximating a polynomial with one with a particular Galois group
Definition: Let $f(X), g(X) \in \mathbb{C}[X]$ be two monic
polynomials of degree $n$. We define a distance (the Hausdorff
distance) between them as follows: Let $\alpha_1, \dots, \alpha_n$ be
the ...
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How does the Galois groups of Taylor polynomials of entire functions relate to the Weierstrass factorization?
How does the Galois groups of Taylor polynomials of entire functions relate to the Weierstrass factorization?
Setup: Let $f$ be a non-polynomial entire function with Weierstrass factorization:
$$f(z)=...
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Divisibility relations among degrees of irreducible factors of a binomial
Suppose $K$ is an algebraic number field, and $a \in K$. Let $n$ be a positive integer. The polynomial $t^n - a \in K[t]$ splits as a product of irreducible factors of degrees $d_1, \dots, d_r$.
Is it ...
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Number field with rational real points
Consider a number field $K$. How to classify or characterize those $K$ intersecting the real line only in the rationals: $K\cap \mathbb{R}=\mathbb{Q}$ ?
An example are quadratic imaginary fields.
In ...
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Finite index subgroups of transcendental Galois groups
Suppose that $K/k$ is a transcendental field extension with both fields algebraically closed (I'm most interested in the case when $\text{char}(k) > 0$). Consider the automorphism group $G = \text{...
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Do you know an (explicit) example of a superelliptic curve $C\!: y^{11} = f(x)$ for which there is a cover $C \to E$ onto an elliptic curve $E$?
Do you know an (explicit) example of a superelliptic curve $C\!: y^{p} = f(x)$ over a field of zero characteristic for which $p = 11$ and there is a cover $\phi\!: C \to E$ onto an elliptic curve $E$? ...
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Is every number field generated by a trinomial?
Can every algebraic number field be constructed by adjoining a root of a trinomial (with rational coefficients) to $\mathbb{Q}$?
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Is there any Galois correspondence about fiber bundle (vector bundle)?
I have learned Galois correspondence about universal covering space over a topological manifold $M$. Since a covering space over $M$ can be viewed as a fiber space over $M$ with discrete fibers, and a ...
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Maximal ideals in rings with polynomial relations
Let $F$ be a perfect field, and let $p(t) \in F[t]$ be an irreducible monic polynomial of degree $n$ such that: $$p(t) = t^n+s_1 t^{n-1}+s_2 t^{n-2}+\dots+s_n$$ Let $\theta_1, \theta_2, \dots, \...