## Separable Extensions (Pt. II)

**Point of Post: **This is a continuation of this post.

From this we can prove the, extremely useful, primitive root theorem. Recall that simple extension is a singly generated one:

**Theorem: ***Let be a finite separable extension, then is simple.*

**Proof: **If is finite then is finite and the conclusion follows from the cyclicity of .

Suppose now that is infinite, so that is infinite. Write as where each is separable. It suffices to show that any separable extension of the form is simple since the general case then follows by induction. So, let be the extension of , as constructed in the previous theorem, with being the map. We then know that there are extensions of to a map . Let and consider . If then we can find two different extensions of the identity map on to a map which are equal on (why?). Note then that evidently otherwise the equation would imply that and thus on (contradictory to assumption). Thus, we see that . Now, since there are only finitely many this clearly implies that there are only finitely many with and so, since is infinite, there exists with as desired.

Since any extension of a characteristic zero field is separable we get the following corollary:

**Corollary:** *Every finite extension of a characteristic zero field is simple. *

In particular, I feel compelled to state the obvious:

**Corollary: ***Every finite extension of is simple.*

Now that we have the primitive element we can state what is, perhaps, the most functional definition of separability of extensions. Just to note this before hand, it involves the tensor products of algebras. Namely:

**Theorem: ***Let be a finite extension. Then, is separable if and only if the -algebra contains no nonzero nilpotents. *

**Proof: **Suppose first that is separable. By the primitive element theorem we can write as for some separable . It is well-known then that we have an -map defined by

with kernel . But, this is actually a map of -algebras and thus we may conclude that, as -algebras,

But, since splits into distinct linear factors we may conclude then by the Chinese remainder theorem that where –and since the product of fields never has any nonzero nilpotents we may conclude that our ring has no non-zero nilpotents.

Conversely, suppose that is not separable. There then exists some non-separable . Note then that since the inclusion gets tensored to an inclusion (since all modules over vector spaces are free, and free modules are flat), and thus if we can show that has a nonzero nilpotent so will . That said, the above analysis shows that, once again, is just as -algebras. But, now we know that for some polynomial . We obviously see then that is non-zero but its square is zero. The conclusion follows.

With this we can easily prove the two following theorems:

**Theorem: ***Let be a finite extension. Then, the extension is separable if and only if each is separable.*

**Proof: **We clearly get the if way for free. Conversely,it suffices to show that contains no nonzero nilpotents. What we claim is that –we prove this by induction. By the previous theorem we can see in particular that if is any field extension with separable then which gives us the base case. For the inductive step we note that

but by inductino hypothesis . Thus,

The theorem then follows since products of fields never have any nonzero nilpotents.

**Theorem: ***Let be a tower of finite fields. Then, is separable if and only if and is separable.*

**Proof: **We have that if and are separable

where we made use of the fact that since is algebraic that . We may clearly then conclude that has no nonzero nilpotents and thus is separable.

Conversely, suppose that is separable. The inclusion implies that has no nilpotents and so is separable. The fact that is separable follows from the fact that in a splitting field.

**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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