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Results tagged with galois-theory
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user 2480
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.
34
votes
1
answer
1k
views
Does any cubic polynomial become reducible through composition with some quadratic?
What I mean to ask is this:
given an irreducible cubic polynomial $P(X)\in \mathbb{Z}[X]$ is there always a quadratic $Q(X)\in \mathbb{Z}[X]$ such that $P(Q)$ is reducible (as a polynomial, and then n …
- 5,103
3
votes
q-th powers and roots of polynomials
A family of counterexamples is defined as follows:
$r=2$,
$p_1\ge 2, \quad p_2\ge 3, \quad d_1 \perp p_1, \quad d_2 \perp p_2$,
$a=\dfrac{\tan(\pi d_1/p_1)-\tan(\pi d_2/p_2)}{\tan(\pi d_1/p_1)+\t …
- 5,103
4
votes
Why do we make such big deal about the 'unsolvability' of the quintic?
Rephrasing in part some of the previous comments and answers, my take is that historically roots, like powers, were seen not as "functions" in the modern sense, but as natural "operations" extending t …
- 5,103