All Questions
Tagged with cubics galois-theory
26
questions
0
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0
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39
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Disjoint exceptional lines on non-minimal cubic surface
A diagonal cubic surface $\sum_{i=0}^3a_iT_i^3=0$ is not minimal if, for example, $a_1a_2a_3^{-1}a_4^{-1}\in(k^*)^3$. This should be because there is an exceptional line $D$ such that no element in ...
- 115
4
votes
3
answers
177
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Prove that $\mathbb Q(\cos\tfrac\pi7)\neq\mathbb Q(\cos\tfrac\pi9)$
Let $\tau=2\pi$ be the full angle. (tau)
For any integer $k$ and any angle $\theta$, $\cos(k\theta)$ is a polynomial in $\cos\theta$. In particular, $\cos(2\theta)=2\cos^2\theta-1$, which shows that $\...
- 5,726
3
votes
4
answers
256
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Prove irreducible cubic polynomial over $\mathbb{Q}$ with a cyclic galois group has real roots
I want to prove the following:
Let $f\in \mathbb{Q}[x]$ be an irreducible cubic polynomial, whose Galois group is cyclic. Prove that all of the roots of $f$ are real.
I know that the Galois group $G$...
- 421
10
votes
0
answers
231
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On the solvable octic $x^8-44x-33 = 0$ and the tribonacci constant
I had discussed the solvable octic trinomial,
$$x^8-44x-33=0\tag1$$
way back in this old MSE post, but I revisited this inspired by another solvable octic,
$$y^8-y^7+29y^2+29=0\tag2$$
which I also ...
- 54.9k
5
votes
0
answers
208
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Factorization of $x^3-x-1$ over $\mathbb{F}_p$
Factoring quadratic polynomials over finite fields can easily be done by determining if the discriminant is a quadratic residue modulo characteristic in question, and if so, apply the quadratic ...
- 405
2
votes
2
answers
101
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Finding the roots of $x^5+x^2-9x+3$
I have to find all the roots of the polynomial $x^5+x^2-9x+3$ over the complex. To start, I used Wolfram to look for a factorization and it is $$x^5+x^2-9x+3=(x^2 + 3) (x^3 - 3 x + 1)$$
I can take it ...
0
votes
1
answer
154
views
What are the necessary and sufficient conditions for the cubic equation to have at least 1 positive real root?
What are the necessary and sufficient conditions for the cubic equation to have at least 1 positive real root?
I'm just dealing with the $2$ simplest cases.
Case-1:
$$x^3+px+q=0$$
where, $p<0$ and ...
- 506
2
votes
1
answer
530
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Denesting Cardano's Formula
For a depressed cubic equation $x^3 + px + q =0$ having exactly one real root, Cardano's formula gives the real root as $$\sqrt[3]{-\frac{q}{2} +\sqrt{\frac{q^2}{4} + \frac{p^3}{27}}} + \sqrt[3]{-\...
- 325
2
votes
0
answers
64
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Galois group of polynomial over $\mathbb{Q}(\sqrt{-3})$
Find the Galois group of the polynomial $f(x) = x^3 -10$ over the field $\mathbb{Q}(\sqrt{-3})$, given that the discriminant of $f$ is $-2700.$
I know that the Galois group of a given polynomial is ...
- 43
2
votes
0
answers
240
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In characteristic 2 the splitting field of a cubic has degree 6
I have been working on the following problem, from an old p-set of a Galois theory course I found online:
Let $F = \mathbb{F}_2(t)$, the field of rational functions on $\mathbb{F}_2$, and let $f(x) = ...
- 895
3
votes
3
answers
571
views
Lagrange Method of Solving Cubic Equations
Let $K$ be a field (for simplicity, one may assume $K\subseteq\mathbb{C}$), and let $d\in K$. Denote $w=\sqrt[3]{d}$, and $\zeta=(-1+\sqrt{-3})/2$. If $f(x)=x^3+ax^2+bx+c$ is the minimal polynomial of ...
- 1,371
1
vote
1
answer
115
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Show that $x^3+3y^3+9z^3-9xyz=1$ has infinitely many integer solutions. [duplicate]
Show that $x^3+3y^3+9z^3-9xyz=1$ has infinitely many integer solutions.
I have found that (1,0,0) and (1,-18,12) are two solutions and tried (1,-18+n,12-n).
There is a hint saying that I should try ...
- 121
1
vote
1
answer
174
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Solving the sextic equation with 14th root of unity
I am solving the sextic equation $t^6-t^5+t^4-t^3+t^2-t+1=0$ satisfied by the 14th root of unity (a problem from Ian Stewart's book). I was able to get up to the point where you have the polynomial $u^...
- 338
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0
answers
71
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Cubic extension adjoining a root of an irreducible cubic polynomial.
Assume $f(x) \in \mathbb{Q}[x]$ is an irreducible cubic polynomial with discriminant $D$, let $x_1, x_2, x_3$ be the three distinct roots of $f(x)=0$. Let $p$ be any odd prime coprime to $D$. Are the ...
- 19
2
votes
2
answers
101
views
None of the methods I know to factor cubics are working here ....
I have been trying to find all the different methods for factoring cubics and so far in my search I have come across:
1)Using the sum/difference of cubes
2)The grouping method
3)Using the rational ...
- 2,815