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. Let be a set in bijective correspondence with the set of all monic irreducible polynomials of degree at least two in . For each let be the corresponding irreducible polynomial. Let be the ring of polynomials with coefficients in in a finite number from a set of variables . We have a ring homomorphism sending an element of to the corresponding constant polynomial.