---- > [!definition] Definition. ([[transcendence basis]]) > > > Let $k \subset L$ be a [[field extension]]. A subset $A$ of $L$ is a **transcendence basis** for $L$ over $k$ if it satisfies one (hence all) of the following equivalent properties: > > 1. $A$ is [[algebraically independent]] over $k$, and $L$ is [[algebraic field extension|algebraic]] over $k(A)$[^1], > 2. $A$ is [[algebraically independent]] over $k$, but $A \cup \{ \beta \}$ is not for any $\beta \in L$, > 3. $L$ is [[algebraic field extension|algebraic]] over $k$, but not over $k(A-\{ \alpha \})$ for any $\alpha \in A$. > > > All transcendence bases for $L$ over $k$ have the same [[cardinality]] (cf. [[transcendence basis#^properties|Property 2]]), called the **transcendence degree of $L$ over $k$** and denoted $\text{trdeg}_{k} L$. If $k \subset L \subset E$, then by [[transcendence basis#^properties|Property 3]] $\text{trdeg}_{k}E=\text{trdeg}_{k}L+\text{trdeg}_{L}E$. > The transcendence degree of an [[integral domain]] is defined to be that of its [[field of fractions]]. > [!intuition] > Compare $(1)$ to the definition of a [[basis]] of a [[vector space]] ([[linearly independent]] + [[submodule generated by a subset|spanning]]), $(2)$ to [[vector space basis iff maximal linearly independent]], and $(3)$ to [[vector space basis iff minimal spanning set]]. ^intuition > [!basicproperties] > 1. If $A \subset L$ is [[algebraically independent]] over $k$, then there is a transcendence basis $B$ for $L$ over $k$ such that $A \subset B$. Putting $A=\emptyset$ witnesses that $L$ has a transcendence basis over $k$. > 2. All transcendence bases for $L$ over $k$ have the same [[cardinality]]. > 3. For [[field|fields]] $k \subset L \subset E$, if $B$ and $C$ are transcendence bases for the [[field extension|extensions]] $L /k$ and $E / L$ respectively, then $B \cup C$ is a transcendence basis for $E / k$. > > Compare $(1)$ [[every linearly independent list extends to a basis]]. Compare $(2)$ to the [[dimension|dimension of a]] [[vector space]] being well-defined. > > > ^properties ---- #### [^1]: The notation $k(A)$ means we are [[ring adjunction|adjoining]] the set $A \subset L$ to the [[ring]] $k$. ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```