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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 $F$.

  1. If $f(x) \in F[x]$, then $f$ splits over $K$ if $f$ factors completely into linear factors in $K[x]$.
  2. 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 $f$.
  3. 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.)

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    $\begingroup$ I normally think this in terms of $F$ and $K$ are sitting in some ambient fixed algebraic closed field. The splitting field really depends on $F\to\bar{F}$ embedding where $\bar{F}$ is the algebraic closure as you know all algebraic closures of $F$ are isomorphic. Once $\bar{F}$ is fixed. Then I would have no trouble to identify the splitting field as there is only 1. $\endgroup$ – user45765 Jun 18 '18 at 20:20
  • $\begingroup$ @user45765 I am tempted to add an edit to the question based on your comment. You said "the algebraic closure" of $F$, whereas Patrick Morandi refers to it using both articles, the as well as a, before and after proving that all algebraic closures (of $F$) are isomorphic. $\endgroup$ – Brahadeesh Jun 18 '18 at 20:24
  • $\begingroup$ @user45765 also, based on your comment, if I always think of a fixed algebraic closure $\bar{F}$ of $F$, then it makes sense to refer to the splitting field $K$, isn't it? That is what I understand from your comment, please correct me if I am wrong. $\endgroup$ – Brahadeesh Jun 18 '18 at 20:26
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    $\begingroup$ Probably the reason is that the author obviously knows the isomorphism theorem and is accustomed to say “the splitting field”, so a couple of slips in the textbook can't be that surprising. Anyway, you're right: before proving that splitting fields are unique up to an isomorphism fixing the base field, one should use “a”. $\endgroup$ – egreg Jun 18 '18 at 21:58
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    $\begingroup$ @Brahadeesh Not necessarily. I'd keep the distinction when clarity requires it. $\endgroup$ – egreg Jun 18 '18 at 22:05
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To summarize a lot of what has been said in the comments (which I tend to agree with), you should use the article "a" before you know the theorem (or if you are talking to someone else who doesn't know the theorem or any context where before we assume we know the theorem, $\dots$). After that, it is uaually safe to refer to "the" splitting field of a polynomial; this frequently happens on this site. However, if there is a need for a distinction between two splitting fields that are not equal (even though they are isomorphic), then we should keep that distinction. But I think such situations are rare.

(Side remark: This reminds me of how in the representation theory textbook by Fulton and Harris, they will put an "$=$" sign for two algebraic structures even when they are only canonically isomorphic but are not equal. I think that the kind of talk we are using here shows up quite a bit in algebra)

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