Let be a field. Suppose is algebraic over , is transcendental over , that is algebraic over , and is transcendental over . Decide whether the following elements are algebraic or transcendental over the given field.
- Is transcendental over ?
Note that is a finite extension of . By Theorem 5.5 in TAN, is algebraic over .
Since and is an algebraic (hence finite) extension of , by Theorem 5.5, is algebraic over .
Suppose is algebraic over . Now is an algebraic, hence finite, extension of . Then is also a finite extension. But , so that (by Theorem 5.5) is algebraic over . So is algebraic over , a contradiction. Thus is transcendental over .
Suppose is algebraic over . Now is a finite extension of , so that is algebraic over – a contradiction. Thus is transcendental over .
Suppose ; then is (trivially) algebraic over . So need not be transcendental over . More generally, consider and .