I am studying from Patrick Morandi's Field and Galois Theory, and in section 3, he makes the following definitions.
Let $K$ be an extension field of $F$.
- If $f(x) \in F[x]$, then $f$ splits over $K$ if $f$ factors completely into linear factors in $K[x]$.
- If $f(x) \in F[x]$, then $K$ is a splitting field of $f$ over $F$ if $f$ splits over $K$ and $K = F(\alpha_1,\dots,\alpha_n)$, where $\alpha_1,\dots,\alpha_n$ are the roots of $f$.
- If $S$ is a set of polynomials over $F$, then $K$ is a splitting field of $S$ over $F$ if each $f \in S$ splits over $K$ and $K = F(X)$, where $X$ is the set of all roots of all $f \in S$.
Often, in mathematics, we take care to use the article the only when the object we are talking about is unique in some sense. This sense is made precise by defining isomorphisms, or homeomorphisms, or any such (bijectively) structure-preserving functions between objects.
So, it makes sense to start out by talking only about a splitting field, because a priori we do not know any two splitting fields of a collection $S$ of polynomials over $F$ are unique up to isomorphism. After such an isomorphism has been shown to exist, we might choose to switch the article and only refer to the splitting field of a collection $S$ of polynomials over $F$.
My problem is that Patrick Morandi freely switches between the two articles when referring to splitting fields, both, before and after proving the Isomorphism Extension Theorem (which proves that splitting fields are unique up to isomorphism). This is a bit jarring for me. I would like to know whether there is any standard regarding which article to use when taling about splitting fields.
(I would love it if the answer is "Yes, use the once you've proved the Isomorphism Extension Theorem, otherwise stick to a." But, I fear that this question is all about nitpicking.)