transcendence degree


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transcendence degree

[tran′sen·dəns di‚grē]
(mathematics)
The transcendence degree of a field E of a subfield F is the number of elements in a transcendence base of E over F. Also known as transcendence dimension.
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Let K be a countable, unbounded, [omega]-homogenous real closed field such that the residue field R of K has infinite transcendence degree over Q.
By Lemma 20 and Lemma 3, it suffices to construct a countable, unbounded, [omega]-homogenous RCF whose residue field has infinite transcendence degree over Q.
There is a transitive model of ZF containing a real closed field K with infinite transcendence degree over Q, but no discrete subset of infinite transcendence degree over Q.
It suffices to observe that discreteness and infinite transcendence degree are the only properties of a residue field section used in the proof of Theorem 21.
An RCF K is supported iff its transcendence degree over Q is finite, i.