Why $p$th power are not allowed in radical extensions?

Question

A field $R$ containing a field $K$ is defined to be radical extension of height $1$ if $R = K(u)$ with $u^p \in K$ for some prime $p$ and $u^p$ is not the $p$-th power of some element in $K$. Then we define the radical extension of height $n$ as a radical extension of height $1$ of a radical extension of height $n-1$.

I don't understand what's the point of adding the stringent condition in the definition of radical extensions. Can anyone provide the motivation for doing so ?

(Other books may follow different definational conventions - so for the record I'm reading Galois theory of algebraic equations by Jean-Pierre Tignol)

Answer 1

If $u^p$ is a $p^\text{th}$ power of some element of $K$ then $K(u) = K$, since you are pretending to extend by something already in $K$. For instance, attempting to extend $\mathbb{Q}$ by a root of $u^2 = 4$ just produces $\mathbb{Q}$.

Answer 2

The idea behind a radical extension is that we're adding a $p$th root for some element of $K$. All that condition does is ensure that the $p$th root we are trying to add isn't already there in some sense.

For instance, let's take our base field to be $\mathbb{R}$. Adding a square root of $-1$ is interesting since the reals don't have square roots of negative numbers, so we get a radical extension $\mathbb{R}[i] = \mathbb{R}[X]/(X^2 + 1)$ (also known as $\mathbb{C})$.

On the other hand trying to add a square root of $2$ isn't interesting. $\mathbb{R}$ already has a square root of $2$. If we try the same construction using polynomials and look at $\mathbb{R}[X] / (X^2 - 2)$, we don't even get a field.

Answer 3

Here's a good reason to make such a restriction. Suppose, after making this definition, you prove that every solvable Galois extension of $\mathbf{Q}$ is a radical extension. Then you deduce that the cyclotomic field $\mathbf{Q}(\zeta_p)$ is also a radical extension since it is abelian. With the weaker notion you are suggesting, it would be an obvious result that $K = \mathbf{Q}(\zeta_p)$ is radical, since $K = \mathbf{Q}(u)$ with $u = \zeta_p$, and $u^p = 1$ is certainly a $p$th power in $\mathbf{Q}$. However, with the stronger definition it is more interesting, and is related to actually expressing $\zeta_p$ as a radical expression in a "more genuine" way, e.g. $\zeta_3 = (-1 + \sqrt{-3})/2$ rather than $1^{1/3}$ and $$\zeta_5 = \frac{1}{4} \left(-1 + \sqrt{5} + \sqrt{-2(5 + \sqrt{5})}\right)$$ rather than $1^{1/5}$.

As a general principle, one reason to make more stringent definitions when weaker definitions are equivalent is proving stronger results in the end.