You should re-read the stuff you think you know.

I started my PhD in June 2011. While I've only worked on a few problems since then, a pattern is emerging: I always spend a couple months stuck on something that looks obvious and silly (both in fore- and hindsight.) Most recently I was stuck on a problem involving basic arithmetic over finite fields. I felt that it was something I should be able to do, but nonetheless was stuck on it for a few weeks.

Maybe it's that I'm doing the PhD thing wrongly, or maybe it's that I'm not spending enough time on my mathematics (I've got a job doing predictive modeling, too!). In any case, the story of how I got unstuck might interest some people:

I've thought it important to continue doing maths reading independent of my research projects. But while there are several subjects I wish to learn about (and do research in), I have not engaged in much outside study. When I approach a subject, I have two separate reactions: Either I find that the subject is too advanced, and I give up quite early - not wanting to get lost in background material. Or I think the material I have repeats too much of stuff I already know, and I give up - not wanting to waste my time on things I've seen before.

After doing this for a few months, I decided that I should read something that I think I already know - just to get in the habit of things. I started reading the field/Galois theory chapters in the abstract algebra book by Dummit and Foote. Pretty light reading as far as mathematics goes. It's stuff I should know.

Those chapters contain some straightforward proofs of basic statements on algebraic field extensions. Exempli gratia, if $\mu$ is the root of a polynomial $f$, and $\sigma$ is an element of the Galois group, then one can prove that $\sigma(\mu)$ is also a root of the polynomial by observing the action of $\sigma$ on the polynomial $f$. Not particularly new or deep material, but it turns out just the stuff I needed...

Now back to my problems with arithmetic over finite fields: I've spent the last few weeks trying to decompose an operator on polynomials into separate components. My supervisor and I both had our own ideas of writing out matrices for these operators. I spent a few late nights on it, wrote down some messy results, and went to bed hoping I'd have an epiphany on how to clean it up. I had been hoping for one for weeks! But soon after I started re-reading the basics I got it. I was just reading about applying the action of the Galois group to polynomials, and here I was staring at an operator on polynomials. So I abandoned the ``matrix approach'', naively started looking at the action of the Galois group, and a few minutes later I had the result I was looking for. It seemed so obvious in hindsight, but it often seems obvious only after you've understood it.

I should probably keep reading the basics.

Cohomology of Finite Fields Part III

I gave a proof (with help from math.stackoverflow) in my last blog for the 0th cohomology of the field extension ${\mathbb{F}_4/\mathbb{F}_2}$. While I have sketches for descriptions of ${H^0(\mathbb{F}_{p^m} / \mathbb{F}_p)}$ for any prime ${p}$ and positive integer ${m}$, I've not yet been able to describe ${H^1}$ for even the simplest cases. I shall describe the problem in this post.

First, recall that I mention in my first post on NCG cohomologies of field extensions that a first order differential calculus gives rise to a differential graded algebra. Let's briefly describe this:

Given our [possibly noncommutative] algebra ${A = \Omega^0}$, we've already described it's first order differential calculus as an ${A}$-bimodule ${\Omega^1}$ together with a linear map ${d: \Omega^0 \rightarrow \Omega^1}$ that obeys the rules 1) ${d(ab) = a \cdot db + da \cdot b}$ (where ${\cdot}$ is the action of the algebra on the bimodule) and 2) ${\Omega^1 = \{a \cdot db : a, b \in \Omega^0\}}$. The differential graded algebra is then defined as graded algebra ${\Omega = \bigoplus_n \Omega^n}$ together with the map ${d : \Omega^n \rightarrow \Omega^{n+1}}$ and the rules: ${\Omega^0 = A}$, ${\Omega^1}$ as before, ${d^2 = 0}$, and ${\Omega}$ generated by ${\Omega^0}$ and ${\Omega^1}$. The product of the differential graded algebra is the wedge product ${\wedge}$, as in classical differential geometry.

Putting the above together, we get a complex ${A = \Omega^0 \xrightarrow{d^0} \Omega^1 \xrightarrow{d^1} \ldots}$. Here ${d^n}$ is the restriction ${d^n = d |_{\Omega^n}}$. The cohomology is then defined as ${H^n(A) = \text{ker}(d^n)/\text{Im}(d^{n-1})}$, with ${H^0(A) = \text{ker}(d^0)}$ as we saw in the last two posts.

In our case of the field extension ${\mathbb{F}_4/\mathbb{F}_2}$, we have as our algebra the polynomial ring ${A = \mathbb{F}_2[x]}$. We've already defined a differential calculus on it, the polynomial ring ${\Omega^1 = \mathbb{F}_4[x]}$, where ${\mathbb{F}_4 = \mathbb{F}_2(\mu)}$, together with the differential map ${df = \frac{f(x + \mu) - f(x)}{\mu}}$. Using this, how can we describe the differential graded algebra ${\Omega = \bigoplus_n \Omega^n}$ ?

First, recall that ${\mathbb{F}_4}$ is a two dimensional vector space over ${\mathbb{F}_2}$ with basis ${\{1, \mu\}}$. This means that the polynomial ring ${\Omega^1}$ can be ``spanned'' by ${\{1, \mu\}}$ over the polynomial ring ${\mathbb{F}_2[x]}$. In other words, every ${f \in \Omega^1}$ can be written as ${f = f_1 \cdot 1 + f_\mu \cdot \mu}$, with ${f_1, f_\mu \in \Omega^0}$. In order for us to get a graded algebra, ${\Omega^2}$ must have a basis ${\{1 \wedge \mu\}}$, hence it's a one dimensional space over ${\Omega^0}$ and thus is isomorphic to ${\Omega^0}$. ${\Omega^n = 0}$ for ${n > 2}$, as we're using a classical anticommutative wedge product.

But what is the map ${d^1: \Omega^1 \rightarrow \Omega^2}$ ? Well, from our description of ${\Omega^1}$, we can easily express the action of ${d}$ on ${\Omega^1}$ as an action on ${\Omega^0}$, which is well defined. Id est, for ${f \in \Omega^1}$,

$\displaystyle d^1 f = d^0(f_1) \wedge 1 + f_1 \wedge d^0(1) + d^0(f_\mu) \wedge \mu + f_\mu \wedge d^0(\mu)$

Now, ${d^0(a) = 0}$ for any constant polynomial, and ${d^0(g) \in \Omega^1}$ for any ${g \in \mathbb{F}_2}$. So ${d^0g = \partial_1 g \cdot 1 + \partial_\mu g \cdot \mu}$. Hence we have linear maps ${\partial_1, \partial_\mu : \mathbb{F}_2[x] \rightarrow \mathbb{F}_2[x]}$. We return to ${d^1}$ using this idea:

$\displaystyle d^1 f = (\partial_1 f_1 \cdot 1 + \partial_\mu f_1 \cdot \mu) \wedge 1 + (\partial_1 f_\mu \cdot 1+ \partial_\mu f_\mu \cdot \mu) \wedge \mu$

The wedge product is anticommutative, so this leaves us with:

$\displaystyle d^1 f = (\partial_1 f_\mu - \partial_\mu f_1) \cdot 1 \wedge \mu$

And ${d^2 f = 0}$, as the rest of the ${\Omega^n}$'s are 0. Now back to the cohomologies: ${H^1(\mathbb{F}_4/\mathbb{F}_2) = \text{ker}(d^1)/\text{Im}(d^0)}$, where ${\text{ker}(d^1) = \{f \in \Omega^1 : \partial_1 f_\mu = \partial_\mu f_1 \}}$ and ${\text{Im}(d^0) = \{f_1 \cdot 1 + f_\mu \cdot \mu \in \Omega^1 : \exists f \in \mathbb{F}_2[x] \; \text{such that} \; df = f_1 \cdot 1 + f_\mu \cdot \mu\}}$. Hence the we need to find a description of the ``partial derivatives'' ${\partial_1}$ and ${\partial_\mu}$. In other words, we need to describe ${d^0f}$ as:

$\displaystyle \frac{f(x + \mu) -f(x)}{\mu} = \partial_1 f + \partial_\mu f \cdot \mu$

By noting that ${\mu^3 = 1}$, I was able to write down a formula for ${\partial_i x^{3n}}$, and then formulas for ${\partial_i x^{3n+1}}$ and ${\partial_i x^{3n+2}}$ in terms of ${\partial_i x^{3n}}$. But the binomial expansion makes these formulas just too messy to work with. I then tried finding a basis under which the maps would have a nice, neat formula, but I had no luck there. Next I tried looking at the action of the Galois group on $d$.

Now, the only non-trivial element of the $\text{Gal}(\mathbb{F}_4/\mathbb{F}_2)$ is the Frobenius automorphism $\sigma(x) = x^2$, exempli gratia, $\sigma\mu=\mu^2=\mu+1$. The action of the Galois group extends to an action on the polynomial ring - one that leaves $\mathbb{F}_2$ fixed. Hence we have

$\displaystyle \sigma d f = \sigma(\partial_1 f) + \sigma(\partial_\mu f \cdot \mu) = \partial_1 f + \partial_\mu f \cdot (\mu + 1) = df + \partial_\mu f$

This allows us to write the ``partial derivatives'' in terms of $d$ and the Frobenius automorphism. Id est,

$\displaystyle $$\partial_\mu f = \sigma (d f) - d f$

and

$\displaystyle \partial_1 f = df - \sigma(d f) \mu + (df) \mu$

It remains to be seen if these formulas for the partials will actually help me calculate the remaining cohomologies.

Cohomology of Finite Fields Part II

Last time we described a theory of cohomology for finite field extensions, but we fell short of calculating ${H^0(\mathbb{F}_4 / \mathbb{F}_2)}$ as I had discovered a mistake in my proof shortly after I had posted the blog. Thanks to the kind folks on math.stackoverflow I have a correct proof (and result!). Let's go through it.

Recall that ${H^0(\mathbb{F}_{4} / \mathbb{F}_2) = \text{ker}(d)}$, where ${\mathbb{F}_{4} = \mathbb{F}_2(\mu)}$, ${\mu}$ a root of a monic irreducible polynomial of degree ${2}$ in ${\mathbb{F}_2}$, and

$\displaystyle d: \mathbb{F}_2[x] \rightarrow \mathbb{F}_{4}[x] \; \text{by} \; df = \frac{f(x + \mu) - f(x)}{\mu}$

Hence ${H^0(\mathbb{F}_{4}/\mathbb{F}_2)}$ is the set of all polynomials ${f \in \mathbb{F}_2[x]}$ such that ${f(x + \mu) = f(x)}$. To find it, first note that if ${f(x) \in H^0}$, then so is ${g(f(x))}$ for any ${g \in \mathbb{F}_2[x]}$, as ${g(f(x + \mu)) = g(f(x))}$. If we can show that some polynomial ${f}$ is the only polynomial in ${H^0}$ with smallest degree, then we know that every other ${g(f(x)) \in H^0}$ is spanned by ${\{f(x)^n : n\in \mathbb{N}\}}$ and further show that every other ${h \in H^0}$ has degree divisible by the degree of ${f}$, then we have that ${\{f(x)^n : n\in \mathbb{N}\}}$ is the basis for ${H^0}$.

Notice also that ${f(x) = x^{4} + x \in H^0}$. This follows from the fact that ${\mathbb{F}_{4}^{\ast}}$ is the cyclic multiplicative group of order ${3}$, hence ${\mu^{3} = 1}$. So we have

$\displaystyle f(x + \mu) = (x + \mu)^{4} + x + \mu = x^{4} + \mu^{4} + x + \mu = x^{4} + x + \mu + \mu = f(x)$

It's not hard to show (exempli gratia, by exhaustion) that ${x^4 + x}$ is the smallest degree polynomial in ${H^0}$, and that it's the only polynomial in ${H^0}$ of degree ${4}$.

Now let ${f \in H^0}$, and let ${\text{deg}f = n}$, say ${f(x) = x^n + a_1 x^{n-1} + \ldots + a_n }$. Clearly ${n}$ cannot be odd, as then ${\binom{n}{1}}$ is also odd, so in the expansion

$\displaystyle f(x + \mu) = (x+\mu)^n + \ldots + a_n = x^n + \mu x^{n-1} + \ldots + \text{lower order terms}$

has a ${\mu}$ as a coefficient of the ${x^{n-1}}$ term, so it can't possibly equal ${f(x)}$.

Now, we want that ${4 | \text{deg}f}$. Say ${\text{deg}f = n = 4k + 2}$. Let's look at the coefficient of ${x^{4k}}$ in ${f(x + \mu)}$. We have ${(x+\mu)^{4k + 2} = (x+\mu)^{4k}(x+\mu)^2 = (x^4 + \mu^4)^k(x^2 + \mu^2) = x^{4k+2} + \mu^2 x^{4k} + \text{lower order terms...}}$. Also ${(x+\mu)^{4k + 1} = (x^4 + \mu^4)^k (x + \mu) = x^{4k+1} + \mu x^{4k} + \text{lower order terms...}}$. Hence in ${f(x + \mu) - f(x)}$, ${x^{4k}}$ has a coefficient ${(\mu^2 + \mu)}$. Thus, if ${f}$ is not divisble by 4, it's not in ${H^0}$.

So say ${f \in H^0}$ has degree ${4k}$. Then ${f(x) - (x^4 + x)^k}$ is also in ${H^0}$ and divisible by 4. In this way we see that polynomials in ${t = x^4 + x}$ form ${H^0}$.