What does it mean for a polynomial to be solvable by radicals?

Question

I am writing a school paper about the Abel-Ruffini theorem. My teacher wants me to explain what it means when a polynomial equation is solvable by radicals. My best guess is that it means that the answer contains a root,

Answer 1

It means that the answer can be computed using only the operations $+$, $-$, $\times$, $\div$, and $n$th roots, in a finite number of steps, using the coefficients of the equation.

Answer 2

Given a finite field extension $F \subseteq K$. A $\textbf{root tower}$ of $K$ over $F$ is a finite sequence of extensions

$$F = K_1 \subseteq K_2 \subseteq \dots \subseteq K_n = K$$

Such that for every $i$ where $1 \le i \le n - 1$, there exists a prime $p_i$ and an element $q_i \in K_{i+1}$ such that $q_i^{p_i} \in K_i$ and $q_i \notin K_i$.

If $f(x) \in F[x]$ where $char F = 0$, we say that the equation $f(x) = 0$ is solvable by radicals, if there exists a finite extension of the splitting field of $f(x)$ that has a root tower over $F$.

Answer 3

You may enjoy reading these books:

• Abel's Proof by Peter Pesic

• Galois Theory for Beginners: A Historical Perspective by Jörg Bewersdorff (MAA review)

and this expository paper:

• Abel and the Insolvability of the Quintic by Jim Brown

See also these answers: 1, 2.

Answer 4

This is the standard dogma about solvability by radicals:

A polynomial is solvable by radicals if every root of the polynomial can be generated from rational numbers using the operations $+,-,\times,\div$, and taking $n$th roots.

For instance, this is a root that can be expressed in terms of radicals: $$ \sqrt[5]{1+\sqrt[4]{\frac{27}{8} + \sqrt 2}} - \sqrt[5]{1+\sqrt[4]{\frac{27}{8} - \sqrt 2}}. $$

I say "dogma" because the definition obfuscates a subtle but important point -- I didn't catch it until I took a class where we proved the Abel-Ruffini theorem. When I say "taking $n$th roots" I mean "taking $n$th roots of a complex number". In general, a complex number has $n$ different roots, and they are trigonometric in nature. For example, the number $1$ has seven $7$th roots, which look like this: $$ 1,\cos\Big(1\cdot\frac{2\pi}{7}\Big)+i\sin\Big(1\cdot\frac{2\pi}{7}\Big),\dots, \cos\Big(6\cdot\frac{2\pi}{7}\Big)+i\sin\Big(6\cdot\frac{2\pi}{7}\Big) $$ In other words, those seven numbers are the roots of the polynomial equation $x^7=1$. According to the standard dogma, those numbers above are "expressible as radicals": indeed, each is $\sqrt[7]{1}$. But this seems wrong according to our intuition since the roots contain trigonometric constants, not radical expressions per se.