Carefully state the following theorems you do not need to pr

Carefully state the following theorems (you do not need to prove them): 1. Theorem concerning the degrees of constructible numbers. 2. Theorem concerning the impossibility of three classically-sought geometric constructions. 3. Theorem concerning the existence and uniqueness of splitting fields.

Solution

1. The Constructible Number Theorem: Every number that you can construct has the following properties:

2. Theorem concerning the impossibility of three classically-sought geometric constructions:

If x is a number obtainable from the rational numbers using only addition, subtraction, multiplication, division, and the taking of square roots, then x is a solution to some polynomial equation with rational coefficients. Moreover, if one factors out irrelevant factors from this equation until one gets down to an \"irreducible\" polynomial equation (one that can\'t be factored any further and still have rational coefficients), the degree of this polynomial will always be a power of 2.

Using this theorem, it is easy to prove the impossibility of the three constructions:

3. Existence and Uniqueness of Splitting Fields:

Let F be a field and let f(x) be a non-constant element of F[x]. Then there exists a splitting field E for f(x) over F.

Splitting Fields are Unique: Let F be a field and let f(x) be a non-constant element of F[x]. Then any two splitting fields of f(x) over F are isomorphic.