Math 361, Spring 2014, Assignment 6
From cartan.math.umb.edu
Carefully define the following terms, then give one example and one non-example of each:[edit]
- Constructible number.
Carefully state the following theorems (you need not prove them):[edit]
- Theorem characterizing the field of constructible numbers.
Solve the following problems:[edit]
- Let F→E be any field extension, and suppose α∈E. Prove that α is algebraic over F if and only if α lies in some finite-dimensional subextension K⊆E.
- Let ¯Q denote the set of all complex numbers that are algebraic over Q. Prove that ¯Q is a subfield of C. (Hint: to show that ¯Q is closed under addition, use the Dimension Formula together with the result of the previous problem. Use similar arguments to demonstrate closure under multiplication and inversion.)
- Prove that ¯Q is an algebraic closure of Q. (Hint: to show that ¯Q is algebraically closed, suppose we have an algebraic extension ¯Q→E and choose α∈E. Put f=irr(α,Q)=a0+a1x+⋯+anxn and let L=Q(a0,a1,…,an,α). Apply a strategy similar to that of the previous problem to the extension Q→L to show that in fact α is algebraic over Q. Now consider the factorization of f over C. Argue that this is in fact a factorization of f over ¯Q. Finally, conclude that α must lie in (the image of) ¯Q.)
- Describe a compass and straightedge construction that produces an equilateral triangle of side length 1/2. Describe the construction informally as a sequence of operations with compass and straightedge, as well as formally as a sequence of points having the properties that we described in class.
Questions:[edit]
Solutions:[edit]
Definitions:[edit]
- Constructible Number.
A real number α is said to be constructible if |α| is the distance between two points occuring in some geometric construction.
Example:
√2 is constructible.
Non-example
3√2 is not constructible.
Theorems:[edit]
- Theorem characterizing the field of constructible numbers.
The set of constructible numbers is a subfield of R. Furthermore, a real number α is constructible if and only if there exists a tower of field extensions Q=E0⊂E1⊂⋯⊂En
with α∈En and, for each i between 1 and n, [Ei:Ei−1]=2.