Abstract Nonsense

Crushing one theorem at a time

Maps of Extensions and the Galois Group


Point of Post: In this post we discuss the morphisms in the “category of all extensions”.

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Motivation

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We have spent a fair amount of time talking about the structures that are field extensions, but we have yet to mention what the “structure preserving maps” between such extensions “should” be. Well, of course, it’s not exactly obvious how to define such maps, but if we stop thinking about an extension k/F as being a field k and a field F with k\supseteq F and instead think about it as an F-algebra k which also happens to be a field then we know what to do. Namely, we already know what the morphisms in F-\textbf{CAlg} (the category of commutative F-algebras) are. Namely, if k,k' are two F-algebras then an F-algebra map f:k\to k' is nothing more than a ring map k\to k' which also respects the F-vector space structure (i.e. f(\alpha k)=\alpha f(k) for all \alpha\in F).

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That said, despite the fact that this is really the reason that we define maps of extensions the way we do, this is rarely said in basic books on the subject. Instead such books like to define a map f:k\to k' of extensions k/F,k'/F to be a ring map such that f_{\mid F}=\text{id}.  Of course, it’s easy to see that such definitions are equivalent. If f is a map of extensions as defined in the first paragraph then necessarily f(\alpha)=\alpha f(1)=\alpha for all \alpha\in F. Conversely, if f is a map of extensions as just defined then f(\alpha x)=f(\alpha)f(x)=\alpha f(x) for all \alpha\in F and x\in k (since f(\alpha)=\alpha for all \alpha\in F by assumption).

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Being a map of extensions is a pretty strong condition. For example, we shall see that if k/F is a finite extension then every map of extensions k\to k is necessarily invertible. Moreover, in general mathematics one can construct a lot of information from an object by examining its automorphism group. This is (hyperbolically) nowhere more true than in the study of field extensions. We shall see that examining the automorphism group of an extension shall enable us to (more so in certain nice cases) read off certain extension-theoretic properties of our extension–this is the so-called field of Galois theory.

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April 24, 2012 Posted by | Algebra, Field Theory | , , , , , | 3 Comments

Another Way of Looking at Induced Representations (Pt. II)


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

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April 24, 2011 Posted by | Algebra, Representation Theory | , , , , , , | 1 Comment

Projections Into the Group Algebra (Pt. I)


Point of post: In this post we discuss the notion of the group algebra, in preparation for our eventual discussion about the representation of symmetric groups.

Motivation

We’ve seen in past posts that the group algebra is isomorphic in all the important ways to the direct sum of matrix algebras. We’ll use this fact to study projections in the group algebra which are functions generalizing the notion of projections on an endomorphism algebra. Namely, projections are elements of the group algebra which are idempotent under convolution. These shall prove to be very important when we attempt, at a later date, to classify the representations of the symmetric group.

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April 9, 2011 Posted by | Algebra, Representation Theory | , , , , , , , , , | 3 Comments

Irreducible Characters (Pt. II)


Point of post: This post is a continuation of this one.

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February 25, 2011 Posted by | Algebra, Representation Theory | , , , , , | 10 Comments

Irreducible Characters


Point of post: In this post we discuss what is arguably one of the most important tools in all of basic representation theory– the irreducible characters of a group.

Motivation

In our last few posts we’ve been developing the notion of a class function and the space of class functions under the hazy motivation that we will use them to ascertain that the cardinality of some set X is the dimension of \text{Cl}(G) (we, in our last post, showed that this was the number k of conjugacy classes in G). In this set we shall see that this set X for which we’d like to show has the quality \#(X)=\dim\text{Cl}(G) are the irreducible characters of the group G. These are certain class functions which will occupy a fair amount of our efforts in the coming posts since, in a very real sense, they make up a vast portion of the substance in basic representation theory. But, for now we shall restrict our attention to defining the irreducible characters and showing that they form an orthonormal basis for \text{Cl}(G). Of this we shall get the corollary that \#\left(\widehat{G}\right)=k.

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February 25, 2011 Posted by | Algebra, Representation Theory | , , , , , | 10 Comments

Class Functions


Point of post: In this post we derive results about the set of class functions on a finite group G, in particular finding its dimension as a subspace of the group algebra and characterizing it as the center of the group algebra.

Motivation

In our last series of posts we saw an interesting technique. We saw the interesting idea that if we want to prove the cardinality of a set X is equal to \kappa it suffices to construct a vector space \mathscr{V} of dimension \kappa such that X is a basis for \mathscr{V}. In particular, we saw that \displaystyle \sum_{\alpha\in\widehat{G}}d_\alpha^2=|G| by considering the group algebra \mathcal{A}(G) of dimension |G| and then showing that \left\{D^{(\alpha)}_{i,j}:\alpha\in\widehat{G}\text{ and }i,j\in[d_\alpha]\right\} is a basis for \mathcal{A}(G) (where, as in the last post, the D^{(\alpha)}_{i,j} are the matrix entry functions). We now wish to get our milage out of this technique by applying it again in a different context (different set and different cardinality). We don’t want to ruin the surprise of what precisely this will be, but we shall construct the vector space with the ‘proper dimension’ in this post. In particular, we will consider and study the set of class functions on a finite group G. Intuitively, these are functions which satisfy (of course we mean this loosely since we haven’t define the domain, range, etc.) the common ‘trace identity’ \text{tr}\left(ABA^{-1}\right)=\text{tr}(B).

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February 24, 2011 Posted by | Algebra, Representation Theory | , , , , | 14 Comments