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Hometranscendence degree

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# transcendence degree

The *transcendence degree* of a set $S$ over a field $K$, denoted $T_{S}$, is the size of the maximal subset $S^{{\prime}}$ of $S$ such that all the elements of $S^{{\prime}}$ are algebraically independent.

The *transcendence degree* of a field extension $L$ over $K$ is the transcendence degree of the minimal subset of $L$ needed to generate $L$ over $K$.

Heuristically speaking, the transcendence degree of a finite set $S$ is obtained by taking the number of elements in the set, subtracting the number of algebraic elements in that set, and then subtracting the number of algebraic relations between distinct pairs of elements in $S$.

###### Example 1 (Computing the Transcendence Degree).

The set $S=\{\sqrt{7},\pi,\pi^{2},e\}$ has transcendence $T_{S}\leq 2$ over $\mathbb{Q}$ since there are four elements, $\sqrt{7}$ is algebraic, and the polynomial $f(x,y)=x^{2}-y$ gives an algebraic dependence between $\pi$ and $\pi^{2}$ (i.e. $(\pi,\pi^{2})$ is a root of $f$), giving $T_{S}\leq 4-1-1=2$. If we assume the conjecture that $e$ and $\pi$ are algebraically independent, then no more dependencies can exist, and we can conclude that, in fact, $T_{S}=2$.

## Mathematics Subject Classification

12F20*no label found*

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## Comments

## "size"

The word size sends to an object from graph theory which is not what you mean. Can you modify it somehow?

Bogdan

## Re: "size"

> The word size sends to an object from graph theory

> which is not what you mean. Can you modify it

> somehow?

Originally, we used the \PMlinkescape commands to

suppress such links, but the current consensus appears

to be that link problems caused by overly common terms

(such as ``normal'', ``size'', or ``regular'') should

be fixed at the source by adjusting link priority.

I've filed a correction to the source entry in this

case:

http://planetmath.org/?op=getobj&from=corrections&id=11759

If you run across similar problems, please consider

filing a similar correction to the source entry.

--

mps

## Re: "size"

mps writes:

> adjusting link priority.

The "priority" directive doesn't have much effect in practice. The "permit" directive is much more useful for reducing bad linkage.

## Re: "size"

Does \PMlinkescape still works? I have tried but it seems that doesnt work or I dont do it correctly...

Thx

## Re: "size"

You have to use either \PMlinkescapeword or \PMlinkescapetext. See this post for more how to use these:

http://planetmath.org/?op=getmsg;id=14716