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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 $\mathbb R$?

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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 $\mathbb R$.

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  • $\begingroup$ So, user12345 is right, and Gallion's definition is incomplete. $\endgroup$ – GEdgar Apr 2 '14 at 13:08
  • $\begingroup$ From now on I'm not trying to remember the numbers, just calling all those guys user12345. If they want to be remembered, they should choose a user name. $\endgroup$ – GEdgar Apr 2 '14 at 13:09

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