Is it justified to say that $\mathbb{R}$ is the splitting field of $x^2-1$ over $\mathbb{R}$?

Question

In Gallian's textbook, he defines a splitting field as follows:

Definition. Let $E$ be an extension field of $F$ and let $f(x) \in F[x]$. We say that $f(x)$ splits in $E$ if $f(x)$ can be factored as a product of linear factors in $E[x]$. We call $E$ a splitting field for $f(x)$ over $F$ if $f(x)$ splits in $E$ but in no proper subfield of $E$.

Take, for example, $x^2-1 \in \mathbb{R}[x]$. This splits over $\mathbb R$ as well as over $\mathbb Q,$ a proper subfield of $\mathbb R.$ Is it justified, then, to say $\mathbb R$ is a splitting field of $x^2-1$ over $\mathbb R$?

Answer 1

Note that $E$ is defined to be an extension of $F$. The definition should really state "no proper subfield of $E$ which contains $F$", because the whole discussion depends on having a constant ground field over which you are working.

Then your statement is OK (as it should be) because $\mathbb Q$ does not contain $\mathbb R$.