It is characterized as the monic polynomial of minimal degree that has u as a root. The degree of the minimal polynomial of u over K is called the degree of u over K. 6.1.5. The next proposition is simply a restatement of Kronecker's theorem. The set of all elements of F that are algebraic over K forms a subfield of F. 6.2.8. An extension field F of K is said to be algebraic over K if each element of F is algebraic over K. 6.2.9.
Every finite extension is an algebraic extension.
Then F is an algebraic extension of K. Geometric constructions 6.3.1. The set of all constructible real numbers is a subfield of the field of all real numbers. Let F be a subfield of R . Let F be a subfield of R . If u is a constructible real number, then u is algebraic over Q , and the degree of its minimal polynomial over Q is a power of 2.