Algebraic independence

Algebraic independence means the absence of a nonzero polynomial relation over the base field. It extends the distinction between algebraic and transcendental elements to finite tuples.

There is a sequence of real numbers for which every finite initial segment consists of algebraically independent elements over the rationals. Given an independent initial segment, specialize each nonzero integer polynomial in one extra variable at those values. At least one coefficient remains nonzero, so each resulting polynomial has finitely many roots. There are countably many such polynomials by countability of integer polynomial rings. Choosing the next real outside the countable union of their roots continues the construction. Clearing denominators makes independence over the integers equivalent to independence over the rationals.

A transcendence basis of a field extension $L/k$ is an algebraically independent set $T$ such that $L/k(T)$ is algebraic. Its cardinality is the transcendence degree. For finitely generated extensions it is finite, and the remaining algebraic extension is finite.

Apply Zorn's lemma to subsets of a field extension with algebraic independence, ordered by inclusion. The union of a chain is independent because any polynomial relation involves only finitely many distinct elements, all of which lie in one member of the chain. A maximal independent set $B$ is a transcendence basis: if any element were transcendental over $K(B)$, adjoining it would preserve independence, by clearing denominators, contradicting maximality. Tuple entries must be distinct in the set-based independence definition; otherwise the nonzero polynomial $X_1-X_2$ vanishes on a repeated pair and excludes every nonempty set.

A transcendence basis $T$ for $L/k$ is separating if $L/k(T)$ is a separable field extension. In this case the $dt$, $t\in T$, form a basis of the Kähler differentials. In characteristic zero every transcendence basis is separating. In positive characteristic the condition on the chosen basis is essential.

For a finitely generated field extension $L/k$ in characteristic zero, a tuple $t_i$ is algebraically independent exactly when its differentials $d{t}_i$ are linearly independent in the Kähler differentials. A relation of minimum total degree gives a nontrivial differential relation; an independent tuple extends to a transcendence basis. In positive characteristic, derivatives of nonconstant relations can vanish identically.

The transcendence degree is the cardinality of a transcendence basis. It counts independent transcendental parameters. In a finitely generated separably generated field extension, it equals the dimension of the Kähler differentials as a vector space over the extension field.

Elements $x_1,\dots ,x_n$ of a field extension of $K$ are algebraically independent over $K$ when no nonzero polynomial in $K[X_1,\dots ,X_n]$ vanishes at $(x_1,\dots ,x_n)$.

The rational function field $K(x_1,\dots ,x_n)$ is the field of fractions of $K[x_1,\dots ,x_n]$ when the variables are algebraically independent over $K$.

Let $t$ be transcendental over a field $k$ and let $\tau$ be the field automorphism sending $t$ to $1/t$. The rational function $u=t+1/t$ is fixed, and $t$ satisfies $X^2-uX+1=0$. Thus the extension over $k(u)$ has degree at most two. It is not one because the nontrivial automorphism $\tau$ fixes $k(u)$. The fixed field of $\tau$ also has index two by the fixed-field theorem, so it equals $k(u)$.

No nonconstant polynomial in $k[t]$ is fixed by inversion. For a polynomial of positive degree, substitution of $1/t$ creates negative powers, whereas the original has positive powers. Uniqueness of the coefficients of a Laurent polynomial rules out equality. Consequently this fixed field cannot have a generator that is a polynomial in the original variable, even though it has the simple rational-function generator above. The statement is valid in characteristic two as well: $t$ and $1/t$ remain distinct transcendental rational functions.