## Separable Extensions (Pt. I)

**Point of Post: **In this post we discuss separable extensions and the separable closure of an extension.

**Motivation**

Last time we discussed normal extensions which were the nice kind of extensions which (among several other definitions) are just the splitting field for a set of polynomials over the ground field. In this post we’d like to discuss another kind of “nice” extension, namely separable extensions. Roughly a separable extension is one where there are multiple roots which become indistinguishable (inseparable, if you will) to the Galois group of the extension. For example, consider the finite field and the rational function field . Then, we claim that is a simple extension of degree . Indeed, it’s a simple extension since . To see that it’s degree it suffices to show that is irreducible (since it annhilates ). To do this we merely note that if factors in it factors in but we already know the irreducible factorization there–. Clearly then (since everything in sight if a UFD) the factorization in must look like for some . But this, in particular, implies that which clearly says that and thus our factorization was trivial. Ok, so what? The important thing to note is that while you can easily verify that is trivial since they are just permutations of the roots of [which, in case this isn’t obvious, there is only one such root].

Similar to the case of we see that the Galois group of our extension has order strictly less than the degree of our extension. The key observation though is that it is for a fundamentally different reason. The reason that is an issue of normality [in the sense of normal extensions]–the polynomial, for which the Galois group is permuting its roots, doesn’t have all of its root in the field. In our case though the polynomial the Galois group is permuting roots of has all of its roots in the field, the problem is that it has repeated roots. The Galois group’s raison d’etre is to permute the roots the best it can, but when there are repeated roots the Galois group has a difficulty doing this because it can’t distinguish (separate) the repeated roots.

Thus, our topic of this thread will be those fields which do not impede the Galois group in doing its job, or at least do not impede it in the way described above.

**Separable Extensions**

We begin with the basic objects of study–separable polynomials. We say that a polynomial is *separable *if it has distinct roots in (the algebraic closure of ). For example, is separable since it has distinct roots in . If a polynomial is not separable we call it *inseparable*.

As defined it’s not totally obvious how to figure out if a given polynomial is separable. Lucky for us though there is a very simple characterization of separable polynomials.

**Theorem: ***Let . Then, is separable if and only if .*

**Proof: **Suppose that . Then, . Suppose first that is separable, then evidently each are distinct and so clearly each appears in all but one term in the expression for and so clearly the two are coprime in and so clearly coprime in Conversely, suppose that . Then, we know we can write

For example, if is characteristic then is not separable since (since ).

In fact, this allows us to conclude the following:

**Theorem: ***Let be irreducible. Then, is separable if and only if . In particular, if is characteristic zero then irreducibles are always separable and if is characteristic then is inseparable if and only if for some . *

We know define the notion of a separable extension. We call a finite extension *separable *if is separable for all . We call an element of the extension *separable *if its minimal polynomial is separable–and thus we see that an extension is separable if and only if all of its members are separable. We obviously define an *inseparable extension *to be a non-separable extension.

For example, any finite extension of (or, in fact, any characteristic zero field) is separable since minimal polynomials are irreducible, and irreducibles are separable.

The extension is not separable since, as we showed in the motivation, which is inseparable since it’s derivative (with respect to !) is zero [or, it’s a polynomial in ].

Now, the first thing I’d like to do is show that a separable extension always has embeddings into some extension . We then use this to show that first non-trivial fact about extensions–the primitive element theorem. The basic idea behind this is that if, for example, is separable, so that is separable, we can find embeddings of into a splitting field of by taking to each of the other roots of . The idea isn’t quite as simple since we only know that a finite separable extensions looks like but the idea is still there:

**Theorem(Primitive Element Theorem): ***Let be a finite separable extension of degree and let be some embedding of fields. Then, there exists an extension such that the number of extensions of to an embedding is .*

**Proof: **We know that for some separable (since everything is separable!) . Let be a splitting field for over . We will show that the number of embeddings extending is . To do this we shall induct on . If we’re done, so assume that it’s true for all extensions with and let . Clearly at least one of the ‘s is not in (otherwise ), we assume that . Now, it’s obvious that the number of extensions of with is just the number of roots of in (this is just the isomorphism extension theorem). Consider now as . Note that since we know, by the inductino hypothesis, that the number of extensions of any given extension is . Since the number of extensions of to is we may conclude that the number of extensions of to is .

**References:**

[1] Morandi, Patrick. *Field and Galois Theory*. New York: Springer, 1996. Print.

[2] Dummit, David Steven., and Richard M. Foote. *Abstract Algebra*. Hoboken, NJ: Wiley, 2004. Print.

[3] Lang, Serge. *Algebra*. New York: Springer, 2002. Print.

[4] Conrad, Keith. *Collected Notes on Field and Galois Theory*. Web. <http://www.math.uconn.edu/~kconrad/blurbs/>.

[5] Clark, Pete. *Field Theory*. Web. <http://math.uga.edu/~pete/FieldTheory.print

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