# Tag Archives: zorn’s lemma

## An application of Zorn’s lemma: Transcendence bases

Zorn’s lemma is a very useful result when it comes to dealing with an infinite collection of things. In ZFC set theory it is equivalent to the well-ordering theorem (every set can be well-ordered) and to the axiom of choice (a Cartesian product of non-empty sets is non-empty). I happened to use it a few days ago in proving the existence of transcendence bases, hence this post!

Zorn’s lemma. If every chain in a poset $P$ has an upper bound in $P$, then $P$ contains a maximal element.

Let $K/k$ be a field extension. We call a subset $S\subseteq K$ algebraically independent if for any $m\ge 0$ and $s_1,\dots,s_m\in S$, $p(s_1,\dots,s_m)=0$ implies $p=0$, where $p\in k[X_1,\dots,X_m]$ is a polynomial. A maximal (with respect to $\subseteq$) algebraically independent subset is called a transcendence base.

Theorem. Every field extension has a transcendence base.

Proof. If $K/k$ is algebraic, the transcendence base is the empty set. Suppose that $K/k$ is not algebraic. Let $\mathcal F$ be the family of all algebraically independent subsets $S\subseteq K$. By Zorn’s lemma, it suffices to show that every chain in $\mathcal F$ has an upper bound. Let $\mathcal C$ be a chain in $\mathcal F$, and let $T=\displaystyle\bigcup_{S\in \mathcal C} S$.

It suffices to show that $T\in\mathcal F$, since then $T$ would be an upper bound for $\mathcal C$ in $\mathcal F$.

Let $P(m)$ be the statement:

If $t_1,\dots,t_m\in T$ are distinct, and $p(t_1,\dots, t_m)=0$ for some $p\in k[X_1,\dots,X_m]$, then $p=0$.

If $t_1\in T$, then $t_1\in S$ for some $S\in \mathcal C$. Since $\mathcal S$ is algebraically independent, $p(t_1)=0$ implies $p=0$ in $k[X_1]$. Thus $P(1)$ is true.

Suppose $P(m-1)$ is true. Let $t_1,\dots,t_m\in T$ be distinct such that $p(t_1,\dots,t_m)=0$ for some $p\in k[X_1,\dots,X_m]$. We can view $p(t_1,\dots,t_m)$ as a polynomial in $t_m$ with coefficients in $k[t_1,\dots,t_{m-1}]$, i.e. say $\displaystyle p(t_1,\dots,t_m)=\sum_{i=0}^n p_i(t_1,\dots,t_{m-1})t_m^i=0$.

Since $P(1)$ is true, it folows that $p_i(t_1,\dots,t_{m-1})=0$ for each $i=0,1,\dots,n$. But then $p_i=0$ for each $i$ by our hypothesis. Thus $p=0$, implying $P(m)$ is true.

Therefore, by induction, $T$ is algebraically independent, i.e. $T\in\mathcal F$. $\square$