## Field Extensions (Pt. II)

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

Let’s now discuss a way of generating (pun intended, as we shall see) new extensions of a given field if we are given an extension. In particular, let be some extension and . Then, by we mean the intersection of all subfields of containing . We can say this differently by defining a *subextension *of to be a field . We see then is the intersection of all subextensions of containing . When we will denote by . For reasons that shall soon become clear it will also behoove us to consider the ring defined to be the intersection of all subrings of containing both and .

It’s not hard to prove that the notation for was not misleading as really is the intersection of all subextensions of (or ) containing . This presentation for (as linear -polynomials in ) is not an accident. We have the following theorem:

**Theorem: ***Let be an extension and . Then, if denotes a set of indeterminates indexed by then the image of the natural map (where denotes the polynomial ring) is . Moreover, is the field of fractions of in .*

**Proof: **Since polynomial rings are the free objects in the set map extends to a -algebra map such that and which is universal with respect to this property. Now, let be any subring of that contains and let be a formal set of variables. We see then that the set map extends to an -algebra map such that . Note then that so that by the universal property of we have that and thus we see that contains . Thus, is a subring of containing and minimal with respect to this property, and thus .

Now, clearly the field of fractions of is a subextension of containing . Moreover, suppose that is another subextension containing . Since is a subring of containing we know that and thus . Thus, is a subextension of containing and is minimal with respect to this property, and so .

Ok, so after writing the above proof I realize it was mostly an exercise in me trying to sound smart. But, that’s how things are proven–right? Regardless, what did it really say? First off, what is the image of the “natural map” ? It’s just -polynomials in the variables . In other words, to get we take every single possible polynomial we can find in and evaluate them at the appropriate values of (e.g. ). So, if this looks like . That’s not so bad, right? What then does look like? Well, since is already sitting inside a field we just have to literally take products of elements of and their inverses, which, in the finite case looks something like

**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.pdf>.

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