The n-Category Café

Suppose $K$ is a field with a subfield $k$. Remember from Part 2 that $x \in K$ is algebraic over $k$ if there’s a nonzero polynomial $P$ with coefficients in $k$ such that $P(x) = 0$. There is then a unique polynomial $P$ with coefficients in $k$ such that

• $P(x) = 0$,
• $P$ has the lowest possible degree for a nonzero polynomial that vanishes on $x$,
• the leading coefficient of $P$ is $1$.

This is called the minimal polynomial of $x$.

If $x \in K$ is algebraic over $k$, we say it’s separable if the derivative of its minimal polynomial is nonzero. And $K$ is a separable extension of $k$ if all its elements are separable.

So much for separable extensions of fields. What about separable algebras over fields? I’ve defined those many times here, but last time I explained the module of Kähler differentials $\Omega^1(A)$ of an algebra $A$ over a field and showed that if $A$ is finite-dimensional, it’s separable iff $\Omega^1(A) \cong 0$. So you can use this fact as our definition for now. Today we’ll put this fact to work!

I got this proof from Tom Leinster during our conversations on this stuff:

Lemma 18. Suppose the field $K$ is a separable extension of the field $k$. Then $\Omega^1(K) \cong 0$.

Proof. It’s enough to show that $d x = 0$ for any $x \in K$.

We know $x$ has a minimal polynomial $p$ over $k$. So $P(x) = 0$, and therefore $d(p(x)) = 0$. Now the usual rules for differentiating sums, products and constants all hold for Kähler differentials, and these imply that $d(P(x)) = P'(x) d x$, where $P'$ is the derivative of $P$ computed using the usual rules. So $P'(x) d x = 0$. By separability, $P'$ is not the zero polynomial. So $P'$ is a nonzero polynomial of smaller degree than $P$, which by minimality implies that $P'(x) \ne 0$. Since $P'(x)$ is an element of the field $K$, it follows that $P'(x)$ is invertible. And since $P'(x) d x = 0$, we have $d x = 0$.    █

Remember that a field $K$ extending $k$ is called a finite extension if it’s finite-dimensional as a vector space over $K$.

Theorem 19. Suppose the field $K$ is a finite separable extension of the field $k$. Then $K$ is a separable algebra over $k$.

Proof. We just saw that $\Omega^1(K) \cong 0$, and in Theorem 14 we saw that any finite-dimensional commutative algebra $A$ over $k$ with $\Omega^1(A) \cong 0$ is separable.    █

We really do need the finiteness condition here! You see, every separable algebra over a field is automatically finite-dimensional, but not every separable field extension is finite. I’ll show you a proof of the first fact someday when I classify separable algebras over a field. For the second fact, we’ll see later today that every extension of $\mathbb{Q}$ is separable — but of course they’re not all finite-dimensional.

Does Theorem 19 have a converse? Yes!

Theorem 20. Suppose the field $K$ is a finite extension of the field $k$. If $K$ is a separable algebra over $k$ then $K$ is a separable extension of $k$.

One way to prove this is by classifying all separable algebras over a field. But there’s also a proof using just Kähler differentials: the idea is to show that if $K$ is a finite extension of $k$ that’s not separable, then $\Omega^1(K) \ncong 0$. After struggling to prove this myself, I gave up and found a proof in a book. I’ll sketch it below.

But first, to build some intuition for this subject, it’s time to look at a field extension that’s not separable!

Inseparable field extensions

Let’s find a field extension that’s not separable, and work out its Kähler differentials. How can we do this? How can we find a field extension containing an element whose minimal polynomial has derivative zero?

Here we are computing derivatives of polynomials ‘formally’, defining

$$\frac{d}{d x} \sum_{i = 0}^n a_i x^i = \sum_{i = 1}^n i a_i x^{i-1}$$

The derivative of a constant polynomial is zero — but a constant polynomial can’t be the minimal polynomial of anything. So how can a nonconstant polynomial have derivative zero?

This can only happen if the coefficients $a_i$ with $i \gt 0$ don’t all vanish, yet still $i a_i$ vanishes for all $i \gt 0$. This can never happen if our field has characteristic zero! In characteristic $p$, it happens when $a_i$ is nonzero only for $i$ that are multiples of $p$.

So, a field of characteristic zero can’t have inseparable extensions. And we only get them in characteristic $p$ when we have an element whose minimal polynomial is of the form

$$P(x) = \sum_{i = 0}^n a_i x^{p i}$$

We could also write this as

$$P(x) = \sum_{i = 0}^n a_i \left(x^p\right)^i$$

And that’s suggestive, because in characteristic $p$ the function $x^p$ is not only multiplicative:

$$(a b)^p = a^p b^p$$

It’s also linear:

$$(a + b)^p = a^p + b^p$$

using the binomial theorem. This weird fact makes algebra in characteristic $p$ really different than in characteristic zero. So this map $x \mapsto x^p$ gets an impressive name: the Frobenius endomorphism. For a field of characteristic $p$ this map clearly has kernel zero, so it’s one-to-one. Thus, for a finite field of characteristic $p$ the Frobenius endomorphism is a bijection.

This implies that we can’t have an inseparable extension of a finite field. Why not? Well, suppose our extension has an element whose minimal polynomial $P$ has $P' = 0$. In characteristic $p$ we’ve seen this means

$$P(x) = \sum_{i = 0}^n a_i x^{p i }$$

But for a finite field we can write $a_i = b_i^p$ since the Frobenius endomorphism is a bijection! So

$$P(x) = \sum_{i = 0}^n \left(b_i x^i\right)^p = \left( \sum_{i=0}^n b_i x^i \right)^p$$

In the second step I used the insane fact that the Frobenius endomorphism is linear. But this means $P$ factors into linear terms, so it can’t have been minimal unless it already were linear, which is also impossible.

We’ve just seen some stuff worth recording:

Theorem 21. Every extension of a field of characteristic zero, or a finite field, is separable.

Since we’re in search of inseparable extensions, this just says where not to look. But that’s still very helpful.

The simplest infinite field of characteristic $p$ is

$$k = \mathbb{F}_p(x)$$

the field of rational functions in $x$ with coefficients in $\mathbb{F}_p$. Let’s try to get an inseparable extension of this!

We need a nonzero polynomial whose derivative vanishes. So use the simplest one:

$$P(x) = x^p$$

Then we need a field extension with an element whose minimal polynomial is $P$. So use the simplest one:

$$K = k[u]/\langle u^p - x \rangle$$

That is, we take $k$ and throw in a new element $u$ whose $p$th power is $x$. This does the job!

What are the Kähler differentials $\Omega^1(K)$ like, thinking of $K$ as an algebra over $k$? Since $d$ of everything in $k$ is automatically zero — these act like ‘constants’ — we mainly need to think about $d u$. The relation

$$u^p = x$$

implies

$$p u^{p-1} d u = d x = 0$$

since $x \in k$. If we weren’t in characteristic $p$, this would force $d u = 0$, as in the proof of Theorem 18. But since we’re in characteristic $p$ this equation is vacuous. So $d u$ obeys no nontrivial relations. Thus $\Omega^1(K)$ is the free $K$-module on one generator, namely $d u$.

Notice: I didn’t prove the last two sentences, but that’s indeed how it works!

Kähler differentials for inseparable field extensions

Now I want to sketch the proof of this:

Theorem 20. Suppose the field $K$ is a finite extension of the field $k$. If $K$ is a separable algebra over $k$ then $K$ is a separable extension of $k$.

This is basically Lemma 1.13(c) in Section 6.1.1 of this book:

• Qing Liu, Algebraic Geometry and Arithmetic Curves, Oxford U. Press, Oxford, 2002.

But let’s see how it goes.

Proof. We’ll prove the contrapositive. We’ll assume $K$ is a finite inseparable extension of $k$, and show $K$ is not a separable algebra over $k$. To do this it’s enough to show $\Omega^1(K) \ncong 0$, since in Theorem 14 we saw that a finite-dimensional commutative algebra $A$ over $k$ is separable iff $\Omega^1(A) \cong 0$.

The simplest case is when $K$ is gotten from $k$ by freely throwing in a single inseparable element $u$ — that is, one whose minimal polynomial has $P' = 0$. Then, as in the example we just looked at, $d u \ne 0$. So $\Omega^1(K) \ncong 0$ in this case.

The only other possibility is that $K$ is bigger. In this case it turns out we can always get $K$ from some field $F$ extending $k$ by throwing in one more inseparable element. So, we have inclusions

$$k \hookrightarrow F \hookrightarrow K$$

Now we have various kinds of Kähler differentials to think about, so we need better notation:

• We have the Kähler differentials of $K$ as an algebra over $k$, which we’ll call $\Omega^1_k(K)$.

• We have the Kähler differentials of $K$ as an algebra over $F$, which we’ll call $\Omega^1_F(K)$.

We’re trying to show $\Omega^1_k(K) \ncong 0$. Since $K$ is gotten from $F$ by throwing in a single inseparable element we have $\Omega^1_F(K) \ncong 0$. So, we’d be done if we could find a surjection $\Omega^1_k(K) \to \Omega^1_F(K)$.

Luckily, this is easy to get. For any commutative algebra $A$ over any commutative ring $R$, $\Omega^1_R(A)$ is the abelian group with generators $d a$ for $a \in A$ and relations

$$d(a + b) = d a + d b , \qquad d(\alpha a) = \alpha d a$$ $$d(a b) = a \, d b + b \, d a , \qquad d 1 = 0$$

for $a, b \in A$ and $\alpha \in R$. So, in our situation $\Omega^1_F(K)$ has the same generators as $\Omega^1_k(K)$ but more relations, namely those of the form $d(\alpha a) = \alpha d a$. This gives our surjection $\Omega^1_k(K) \to \Omega^1_F(K)$, as desired.    █

In fact, for any homomorphisms of commutative rings

$$k \to F \to K$$

there is an exact sequence

$$\Omega^1_k(F) \otimes_F K \to \Omega^1_k(K) \to \Omega^1_F(K) \to 0$$

so $\Omega^1_k(K)$ maps surjectively onto $\Omega^1_F(K)$. I won’t explain this exact sequence here, but you can find it in many treatments of Kähler differentials. In Qing Liu’s book it’s Proposition 1.8 of Section 6.1.1. In the Stacks Project it’s Lemma 10.131.7. In Matsumura’s Commutative Algebra it’s Theorem 57. In my favorite book on separable algebras:

• Timothy J. Ford, Separable Algebras, American Mathematical Society, Providence, 2017.

it’s called the First Fundamental Exact Sequence, and it’s constructed in Theorem 8.3.3. So, everyone who knows their Kähler differentials knows and loves this exact sequence!