Year of Short Fiction Part 3: The Red Pony

The Red Pony is in one sense a novella published by John Steinbeck, but in another sense it is a collection of four short stories, originally published episodically in magazines. This makes it slightly difficult to pin down exactly what date to slap on this. The first story was published in 1933, so it came out before all his most famous works.

I was pretty excited to read this, because Steinbeck is one of the best long-form “family epic” writers. His masterpiece, in my eyes, is East of Eden, which chronicles several generations in great detail. It is true that The Pearl and Of Mice and Men and other of his short works pack a punch, but nothing compares to the deep characterization he pulls of in his longer works.

I’m torn on this one. It’s certainly my least favorite of the short fiction series so far. I can intellectualize it’s strong points, but I didn’t connect with any part of it. And the end is super weird, but we’ll get to that. Obviously there will be “spoilers,” but I haven’t really been saying that considering these stories are a hundred years old and only take an hour or so to read.

All four stories form key moments in Jody’s maturation from childhood to adulthood. Steinbeck does a great job of establishing his innocence in a small amount of space by dropping small details throughout the beginning. One of the most interesting was that Jody had a rifle, but he wasn’t allowed bullets until he demonstrated maturity with it for a full year.

This is Steinbeck establishing family dynamics and rituals. It shows that Jody hasn’t undergone one of the key rituals on the path to adulthood. The first story is about how Jody trusts Billy to take care of a horse that eventually dies. Steinbeck cleverly foreshadows this by mentioning the buzzards at the start that eventually deal the horse’s final blow.

I see the novella as a sequence of four deaths and how Jody matures in reaction to each as he ages. In reaction to the death in the first story, Jody lashes out in anger and can’t understand how the adults in his life didn’t protect him from it.

The second death is stranger. A man comes to the family’s house and wants to live out his last days there because it is where he was born. This brings another perspective to Jody. The man rides off to the mountains with an old, dying horse. Again, the horse and human presumably die, but Jody learns of a more mature way to accept the inevitability of death through this stranger.

The third death is of a pregnant horse. This horse must die to save the pony inside of it. This teaches Jody of the circle of life. Death can bring about new life, which itself will eventually die.

Though Jody doesn’t realize it, this is a redemption story for Billy. Billy had promised not to let the horse die in the first story, and he feels guilt for letting it happen. Here, he promises Jody the colt inside the horse, and he has to kill the horse to deliver it. He succeeds in his promise this time. He gives Jody a horse to make up for the one that died and can let go of his guilt.

The final story doesn’t actually have a death in it, but Jody’s grandfather comes to live with the family. The grandfather participated in traveling west across the country. It was a grand adventure, but the grandfather makes it very clear such adventure is over. This kills Jody’s dreams of doing the same.

This is the final straw in Jody’s maturation. He learned of death, life, violence, the fallibility of adults, and now his boyhood dreams are gone. He must learn to live pragmatically in the real world.

The ending was extremely strange at first.

Jody ran into the kitchen where his mother was wiping the last of the breakfast dishes. “Can I have a lemon to make a lemonade for Grandfather?”

His mother mimicked—“And another lemon to make a lemonade for you.”

“No, ma’am. I don’t want one.”

“Jody! You’re sick!” Then she stopped suddenly. “Take a lemon out of the cooler,” she said softly. “Here, I’ll reach the squeezer down to you.”

Lemonade hadn’t made an appearance for the whole novella. What on Earth could this ending be?

It solidifies the idea that Jody has fully matured. His youthful self merely would have feigned interest in helping his grandfather to get himself a lemonade. The only reason the mother can think of that he wouldn’t want one is that he’s sick. But then she realizes he has matured. He’s acting selflessly, and so she encourages it.

I get what Steinbeck was doing. I just didn’t find it very compelling. I dreaded picking it up when I stopped between stories. There is pretty much no narrative momentum. Part of this comes from the stories being early in Steinbeck’s career, but I think when you look at it broken down in the way I did, it becomes clear that this is first and foremost a carefully constructed exercise. It’s obviously well done. I just didn’t like it much.

Local Class Field Theory

Today will probably be our last class field theory post. I want to end with a brief description of local class field theory. Let {K} be a global field, and {v\in M_K} a place. We have our standard inclusion {i_v: K_v^*\hookrightarrow \mathcal{J}_K} by putting the element in the {v} component and {1}‘s everywhere else. Suppose {L/K} is abelian. We have the Artin map {\psi_{L/K}: C_K/N_{L/K}(C_L)\stackrel{\sim}{\rightarrow} G=Gal(L/K)}.

What we learned two posts ago is that the image upon composing the maps {K_v\rightarrow \mathcal{J}_K\twoheadrightarrow C_K/N_{L/K}(C_L)\rightarrow G} is exactly the decomposition group {G_w} where {w} lies over {v}. The image of the units {\mathcal{O}_v^*} under the map is the inertia group {I_w}. This gives us two exact sequences that fit together:

\displaystyle \begin{matrix} 1 & \rightarrow & N_{L_w/K_v}(L_w^*) & \rightarrow & K_v^* & \rightarrow & G_w & \rightarrow & 1 \\ & & \cup & & \cup & & & & \\ 1 & \rightarrow & N_{L_w/K_v}(\mathcal{O}_w^*) & \rightarrow & \mathcal{O}_v^* & \rightarrow & I_w & \rightarrow & 1\end{matrix}

This gives us a local Artin map {\psi_{w/v}: K_v^*\rightarrow G_w}. The main theorem of local class field theory is that the map is surjective with kernel {N_{L_w/K_v}(L_w^*)} and moreover the map can be defined independently of localizing the global fields. Just like global class field theory there is an “existence” part of the theorem as well.
This part says that every finite abelian extension of local fields arises as the localization of an extension of global fields and the local Artin maps {\psi_{w/v}} give a bijection between

\displaystyle \left\{ \text{finite index open subgroups in} \ K_v^*\right\} \leftrightarrow \left\{\text{finite abelian ext} \ L_w/K_v \right\}

where the correspondence is {U \leftrightarrow \ker \psi_{w/v}}.
Let’s do the simplest example. Let’s think about the quadratic extensions {\mathbb{Q}_p} by looking at the bijection. The standard construction is that {\mathbb{Q}_p(\sqrt{d})} are the quadratic extensions where {d} is not a square, and {\mathbb{Q}_p(\sqrt{d'})} is the same extension if {d/d'} is a square. Thus there is a nice bijection between the quadratic extensions and the non-trivial elements of {\mathbb{Q}_p^*/(\mathbb{Q}_p^*)^2}.

Local class field theory tells us that the quadradic extensions of {\mathbb{Q}_p} are in bijection with the open index {2} subgroups {U\subset \mathbb{Q}_p^*} via {L/\mathbb{Q}_p\mapsto N_{L/\mathbb{Q}_p}(L^*)}. It turns out that any index {2} subgroup at all must be open because it will contain {(\mathbb{Z}_p^*)^2}.

Now it will be useful to think in different terms, which is actually a more standard modern reformulation of class field theory since it generalizes to “higher dimensions.” There is a bijection between the open index {2} subgroups {U} of {\mathbb{Q}_p^*} and surjective characters {\chi: \mathbb{Q}_p^*\rightarrow \{\pm 1\}} just by taking the kernel of the character. Thus we have reformulated the problem of counting these subgroups to counting characters.

The unit groups have the form (if {p} odd) {\mathbb{Q}_p^*\simeq p^\mathbb{Z}\times \mathbb{Z}_p^*} and {\mathbb{Z}_p^*\simeq \mu_{p-1}\times (1 + p\mathbb{Z}_p)} where {\mu_{p-1}} is thought of via the Teichmuller lift. If {\chi} has order {2}, then it is trivial on {(\mathbb{Q}_p^*)^2\simeq p^{2\mathbb{Z}}\times (\mathbb{Z}_p^*)^2\simeq p^{2\mathbb{Z}}\times (\mu_{p-1})^2\times (1+p\mathbb{Z}_p)}.

Thus we get the result that {\chi} factors through {p^\mathbb{Z}/p^{2\mathbb{Z}}\times \mu_{p-1}/(\mu_{p-1})^2} which is a finite abelian {2}-group of order {4}. Thus the number of non-trivial characters is {3}. This gives us a nice alternate description to the classical Kummer description. It tells us there are exactly {3} quadratic extensions up to isomorphism. These aren’t hard to figure out explicitly either. Just take some {a\in \mathbb{Z}_p^*} which is not a square. The three quadratic extensions are {\mathbb{Q}_p(\sqrt{p})}, {\mathbb{Q}_p(\sqrt{a})}, and {\mathbb{Q}_p(\sqrt{ap})}. A similar description can be computed when {p=2}, but you get {7} in that case.

That is all for class field theory for now. We’ll move on to complex multiplication, and I think we’ve done enough of the basics that we can probably do what we need as we need it now.

Irreducible Character Basis

I’d just like to expand a little on the topic of the irreducible characters being a basis for the class functions of a group cf(G) from two times ago.

Let’s put an inner product on cf(G). Suppose \alpha, \beta \in cf(G). Then define \displaystyle \langle \alpha, \beta \rangle =\frac{1}{|G|}\sum_{g\in G} \alpha(g)\overline{\beta(g)}.

The proof of the day is that the irreducible characters actually form an orthonormal basis of cf(G) with respect to this inner product.

Let e_i=\sum_{g\in G} a_{ig}g. Then we have that a_{ig}=\frac{n_i\chi_i(g^{-1})}{|G|} (although just a straightforward calculation, it is not all that short, so we’ll skip it for now). Thus e_j=\frac{1}{|G|}\sum n_j\chi_j(g^{-1})g.

So now examine \frac{\chi_i(e_j)}{n_j}=\frac{1}{|G|}\sum \chi_j(g^{-1})\chi_i(g)
=\frac{1}{|G|}\sum \chi_i(g)\overline{\chi_j(g)}
= \langle \chi_i, \chi_j \rangle.

Where we note that since \chi_j is a character \chi_j(g^{-1})=\overline{\chi_j(g)}. Thus we have that \langle \chi_i, \chi_j \rangle = \delta_{ij}.

This fact can be used to get some neat results about the character table of a group, and as consequences of those we get new ways to prove lots of familiar things, like |G|=\sum n_i^2 where the n_i are the degrees of the characters. You also get a new proof of Burnside’s Lemma. I’m not very interested in any of these things, though.

I may move on to induced representations and induced characters. I may think of something entirely new to start in on. I haven’t decided yet.