Splitting field notation confusion

Question

I'm seriously having trouble in understanding the notation used for a splitting field:

Let $f\in K[t]$. Is the splitting field notation $Σf/K$ or just simply $Σf$? What is the difference between the two?

For example, let $t^2 + 1\in \mathbb{Q}[t]$. I know the splitting field of this polynomial is $\mathbb{Q}(i)$; does this mean $Σf/\mathbb{Q}=\mathbb{Q}(i)$ or just simply $Σf=\mathbb{Q}(i)$?

I believe it should be the latter since what I currently understand is that the notation $Σf/\mathbb{Q}$ simply means there exists an injective ring homomorphism between the fields $Σf$ and $\mathbb{Q}$, and so it doesn't actually equal a set.

Answer 1

Both notations are the same, in $\Sigma_f/K$ the $/K$ part is here to emphasize on the fact that one take a splitting field of a polynomial over a given field which contains the polynomial coefficients.

It makes no sense to talk about a splitting field of a polynomial without telling over which field. For example, a splitting field of $f:=x^2-2$ over $\mathbb{Q}$ is $\mathbb{Q}(\sqrt{2})$, whereas over $\mathbb{R}$ it is $\mathbb{R}$. With your notations, one has: $$\Sigma_f/\mathbb{Q}=\mathbb{Q}(\sqrt{2})\neq\mathbb{R}=\Sigma_f/\mathbb{R}.$$

As a side note, do not talk about the splitting field of a polynomial over a given field $K$, it is unique only up to $K$-isomorphims.

Answer 2

I don't know this notation, but it has to be related to the field of coefficients, unless the context makes it obvious.

For instance, as you wrote, for $f(x)=x^2+1\in\mathbf Q(x)$, one has $\;\Sigma f/\mathbf Q\simeq\mathbf Q(i)$.

But if $f(x)=x^2+1\in\mathbf F_2(x)$, one has $\;\Sigma f/\mathbf F_2\simeq\mathbf F_2$ since in any field of characteristic $2$, $x^2+1=(x+1)^2$.

In $\mathbf F_3$, the polynomial $x^2+1$ is irreducible since it has no root, and its splitting field is isomorphic to $\mathbf F_3(x)/(x^2+1)\simeq \mathbf F_9$, the field with $9$ elements.