Facts
If M/L is a field extension and L /K is another field extension, then the transcendence degree of M/K is equal to the sum of the transcendence degrees of M/L and L/K. This is proven by showing that a transcendence basis of M/K can be obtained by taking the union of a transcendence basis of M/L and one of L /K.
Read more about this topic: Transcendence Degree
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