An extension field of a field F that is not algebraic over F, i.e., an extension field that has at least one element that is transcendental over F. For example, the field of rational functions F(x) in the variable x is a transcendental extension of F since x is transcendental over F. The field R of real numbers is a transcendental extension of the field Q of rational numbers, since π is transcendental over Q.
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