All Questions
Tagged with cubics abstract-algebra
25
questions
3
votes
1
answer
139
views
Solving a depressed cubic polynomial in modulus. [closed]
Is there a general technique for solving depressed cubic modulus polynomial? For instance, how would you solve the equation $a^3 + a + 21 = 0 \pmod{43}$?. My attempts eventually ended up with solving $...
4
votes
2
answers
492
views
Understanding Cardano's Formula
In deriving his formula, Cardano arrives at the equation $y^3+py+q=0$. By substituting $y=\sqrt[3]{u}+\sqrt[3]{v}$, he gets the equation $(u+v+q)+(\sqrt[3]{u}\sqrt[3]{v})(3\sqrt[3]{u} \sqrt[3]{v} +p)=...
- 1,467
5
votes
0
answers
109
views
Counterexample to the Hasse principle of the form $x^3 + y^3 + z^3 + nt^3$
Selmer's cubic is a counterexample to the Hasse principle for ternary cubic forms.
We also know that the Hasse principle does not hold for quaternary cubic forms, as
$$
5x^3 + 12y^3 + 9z^3 + 10t^3
$$
...
- 2,907
5
votes
0
answers
714
views
Elementary approach to solving cubic equations over finite fields
I am interested in studying cubic equations over finite fields. For example, when does
$$
ax^3 + bx^2 + cx + d = 0
$$
have a solution in $\mathbb{F}_q$ for $a,b,c,d\in \mathbb{F}_q$ (finite field of ...
- 3,309
2
votes
1
answer
182
views
Proving the sums of three cubes conjecture by the Hasse principle
In his Cours d'arithmétique Serre applies the Hasse-Minkowski theorem to quadratic forms of the form:
$$
x^2 + y^2 + z^2 = n
$$
for $n \in \mathbb{N}$ to prove that a natural number $n$ is a square if ...
- 2,907
0
votes
3
answers
2k
views
Solution of cubic equations in terms of quadratic equations
Is there a general way in which a cubic equation of $3$rd degree can be represented by a quadratic equation of $2$nd degree such that $2$ solutions of a cubic equation is equal to $2$ solutions of ...
0
votes
1
answer
65
views
How to get rid of such radicals?
I would like to know if there is any way I can get rid of these cubic radicals bellow (1). I am allowing both complex and real values.
$$ \sqrt[3]{ -\frac{q}{2} + \sqrt{\frac{q^2}{4} + \frac{p^3}{27} }...
- 516
5
votes
3
answers
154
views
Using partial information to factor $x^6+3x^5+5x^4+10x^3+13x^2+4x+1.$
I wish to find exact expressions for all roots of $p(x)=x^6+3x^5+5x^4+10x^3+13x^2+4x+1.$ By observing that for the roots $x_0 \pm iy_0, x_0 \approx -0.15883609808599033632, y_0 \approx 0....
- 5,230
1
vote
1
answer
174
views
Solving the sextic equation with 14th root of unity
I am solving the sextic equation $t^6-t^5+t^4-t^3+t^2-t+1=0$ satisfied by the 14th root of unity (a problem from Ian Stewart's book). I was able to get up to the point where you have the polynomial $u^...
- 338
2
votes
3
answers
1k
views
Prove $2\cos\frac{2\pi}{7}$ is root of $x^3 + x^2 - 2x -1$
To prove that a regular septagon cannot be constructed by a straightedge and compass, it suffices to prove that $2\cos(\frac{2\pi}{7})$ is not constructible.
Several other answers to this question ...
- 535
1
vote
2
answers
64
views
how one can prove that real roots, which look non-real when using Cardano's formula.
how one can prove that real roots, which look non-real when using Cardano's formula, can be shown to be real.
- 11
1
vote
2
answers
131
views
Writing Cubic Equation in terms of discriminant (with possible shifts and translations)
So I noticed this fact for the following fact for quadratic equations. I need one notation that if one equation can be gotten from another through a shift or scaling of variable then I will denote ...
- 1,625
5
votes
1
answer
337
views
For $c\in\mathbb{F}_p^*$, the cubic $t^3-3ct^2-3t+c$ has exactly one root $r\in\mathbb{F}_p$. Express $r$ in terms of $c$ without cubic roots.
For some $c \in \mathbb{F}_p^*$ consider the polynomial
$$
f(t) = t^3 - 3ct^2 - 3t + c
$$
for $p \equiv 1$ (mod $3$) and $p \equiv 3$ (mod $4$). In this case $3$ is a quadratic non-residue modulo $p$ ...
- 243
1
vote
0
answers
39
views
Solutions of Cubic equation over $k$ with $char(k)=0$
I was working on the irreducibility of cubic equations over a non specific field (at first) and came up with this question: Given a cubic polynomial $d(t)=t^3+at^2+bt+c$ in $k[x]$ where $char(k)=0$, ...
- 893
-2
votes
1
answer
77
views
How to find $\operatorname{Gal}(S/\mathbb{Q})$ [closed]
Please help me to answer the following problem:
Let $f(x)=x^3-3x-5\in\mathbb{Q}[x]$
Thanks
- 291