Point of Post: In this post we discuss the notion of field extensions and plenty of examples.
I’ve been a little behind with blogging recently, and since I’ve been doing a lot of Galois theory recently I thought I’d brush up on some of the basics by blogging about it. In this post we are going to discuss the fundamental objects in Galois theory–field extensions. Historically field extensions arose from the desire to extend a given field to obtain the roots of a (some) given polynomial(s). For example, the complex numbers could be thought of (algebraically) as mathematician’s answer to the difficulty of the equation not having any solutions in the field . Indeed, one can algebraically think about (as a ring) as nothing more than which can basically be described as taking and appending a formal variable to get a ring subject (besides the obvious ones) to the condition that . Of course we have a different name for this , we call it , and we call the result of appending with the field of complex numbers. More generally one may ask whether given a field and some polynomial where we can find some “larger” field such that the polynomial has a root, or more brashly, can be factored into linear terms. The reason for this, besides pure intellectual curiosity, can be seen in the utility of using in the study of the algebra of and . Indeed, many questions dealing exclusively with the real numbers or real polynomials can be reformulated into easier to understand, and often times, easier to solve problems involving . Just a small taste of this can be glimpsed by the notion of divisibility in . For example It’s hard for me to tell if divides just by looking at it, but luckily I can quickly check that this latter polynomial has as a root in from where (by general nonsense) I know that does indeed divide it. Of course, this is a paltry use of extending fields compared to the use that’s touted as one of the most beautiful results in mathematics (at least to mathematicians)–the insolvability of the general quintic. Roughly this “insolvability” says that the notion of a “quadratic equation” (or cubic, or quartic) does not exist for quintic equations. Why is it feasible that this problem should involve field extensions? What the problem is really stating is that we can’t start with the coefficients of our polynomial with (say) integer coefficients and express the roots of the polynomial by performing a finite number of sums, multiplications, and taking of roots (e.g. ). The statement is a non-existence theorem involving rational numbers and roots, but more importantly it’s discussing the roots of a certain polynomial. Thus, if we’d like to make any sort of formal workings with these roots we better have a field that contains them–thus field extensions.
Remark: For all the posts on field theory/Galois theory we shall be working in which is to say that our rings will have units, and more importantly, our maps need to respect (i.e. under any ring map).
Let’s begin by defining exactly what we mean by “field extension”. Let be some given field, then an extension of or a field extension over is a field along with an embedding (of rings) . Note though that the quantifier embedding is really quite unnecessary since any ring map out of field is necessarily injective. Thus, a field extension of is nothing but a field along with a distinguished ring map . In this way we see that field extension of are nothing more than -algebras where we require the algebra to actually be a field.
Let’s look at some examples:
The field can be thought of as an extension over with the inclusion map .
The field is readily verified to be a field and can then be thought of as an extension of with the inclusion mapping.
More exotically, let denote the (open) unit disc then the ring of all meromorphic functions on is readily verified to be a field, which can be thought of as an extension of by the map sending a complex number to its associated constant function.
Let be integral domains then (where denotes field of fractions) and can be thought of as extension of by inclusion. For example, applying this to gives our second example. Also, applying this to (the ring of holomorphic functions on ) gives our third example. Another example of this is thinking of , where is some field, which tells us that is an extension where is the rational function field over (i.e. the quotient of polynomials in with nonzero denominator).
Let be a field and let be irreducible. We know then that is a maximal ideal and thus is a field, which can be thought of as an extension of via the map sending . In fact, this also generalizes our second example since, as one can check, .
Because the faithful way in which a field is put into an extension we often times indulge a (very sweet) logical inaccuracy by identifying with its image so that we literally think of as sitting inside of . Thus, one would not lose much by defining a field extension of to be a field with . With this map-free point of view we use the notation to denote that is an extension of (note there is not quotienting here–it’s just a convenient notation reminding us that is over ).
Let’s discuss one more important example of field extensions. Namely, let be any field. Recall that is an initial object in , or in much less obnoxious language, the fact that is a ring tells us that there is a unique ring map . By studying the image and kernel of this map we were led to the notion of prime subring and characteristic of a ring. While characteristic still works well with fields the notion of prime subring needs some improvement. Indeed, for this map we have two choices–either the map has kernel for some prime or it’s injective corresponding to the possibilities and repsectively (this follows from the more general fact that for an integral domain, the characteristic is either prime or zero). Now if then by the first isomorphism theorem we know that we get an embedding and thus we can think of as an extension of . Now, if is injective we know by the universal characterization of quotient fields that our ring map lifts to a ring map which (by necessity) is an embedding, and thus we can think of as an extension of .
Summarizing the above we see that given a field we can think of as being an extension of or according to the two cases and respectively. In this context we call the or the prime subfield of . Intuitively it is the smallest subfield of . Formally, it’s the smallest subfield of . You can get it by just intersecting all the subfields of , or more constructively you can take the smallest unital subring of , i.e. and look at it’s field of fractions inside . We often times denote the prime subfield of by .
 Morandi, Patrick. Field and Galois Theory. New York: Springer, 1996. Print.
 Dummit, David Steven., and Richard M. Foote. Abstract Algebra. Hoboken, NJ: Wiley, 2004. Print.
 Lang, Serge. Algebra. New York: Springer, 2002. Print.
 Conrad, Keith. Collected Notes on Field and Galois Theory. Web. <http://www.math.uconn.edu/~kconrad/blurbs/>.
 Clark, Pete. Field Theory. Web. <http://math.uga.edu/~pete/FieldTheory.pdf>.