Tagged Questions
4
votes
4answers
192 views
Prove $\sqrt[3]{3} \notin \mathbb{Q}(\sqrt[3]{2})$
I've tried solving $\sqrt[3]{3} =a + b* \sqrt[3]{2}+c* \sqrt[3]{4}$, but there is no obvious contradiction, even when taking the norms/traces of both sides. I can't think of another approach. This is ...
1
vote
2answers
48 views
Larger Theory for root formula
Consider the quadratic equation:
$$ax^2 + bx + c = 0$$
and the linear equation:
$$bx + c = 0$$.
We note the solution of the linear equation is
$$x = -\frac{c}{b}.$$
We note the solution of the ...
7
votes
1answer
137 views
closed-form expression for roots of a polynomial
It is often said colloquially that the roots of a general polynomial of degree $5$ or higher have "no closed-form formula," but the Abel-Ruffini theorem only proves nonexistence of algebraic ...
1
vote
2answers
145 views
Irreducible polynomials with integer coefficients over Q
Suppose p(x) is an irreducible polynomial over Q of degree n, with integer coefficients. If p(x) has two roots r1 and r2 satisfying r1r2 = 5, prove that n is even.
Attempt at solution:
Because the ...
2
votes
3answers
267 views
Algebraic conjugates
Suppose $L/K$ is an algebraic field extension. Take $\alpha_1 \in L$. Then $\alpha_1$ has minimal polynomial $f(x)$ over $K$. Let $\alpha_2, ... \alpha_k$ be the other roots of $f$ in $L$. The ...
23
votes
3answers
622 views
Galois groups of polynomials and explicit equations for the roots
Lets say I have calculated the galois group of some polynomial and I also have the subgroup structure. What's an effective procedure to turn the group into equations for the actual roots of the ...
