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. It is impossible to find a general construction for trisecting an angle, duplicating a cube, or squaring a circle. Splitting fields 6.4.1.
L be an isomorphism of fields.
Then F is the splitting field of the polynomial x k -x over the prime subfield of F. Example 6.5.1. Two finite fields are isomorphic if and only if they have the same number of elements. Let F be a field with p n elements. Each subfield of F has p m elements for some divisor m of n. Conversely, for each positive divisor m of n there exists a unique subfield of F with p m elements.