Each element of is algebraic over . Let be the coefficients of the minimal polynomial over of . If we can prove that the extension is finite, it will follow that its intermediate extension is finite. For the inductive step, we note that is finite, because the minimal polynomial of over has degree less than or equal to the minimal polynomial of over , and that the composite extension is finite because of the inductive hypothesis and the multiplication theorem. Every polynomial in of degree greater than zero has a root in .
Every algebraic extension is an isomorphism.
For any extension and element , if then is transcendental over . A polynomial in is irreducible if and only if it has degree one. We say that an extension is an algebraic closure of if is algebraically closed, and every extension with algebraically closed has as an intermediate extension. Every field has an algebraic closure. Let be a field.