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]

  1. Constructible number.

Carefully state the following theorems (you need not prove them):[edit]

  1. Theorem characterizing the field of constructible numbers.

Solve the following problems:[edit]

  1. Let FE be any field extension, and suppose αE. Prove that α is algebraic over F if and only if α lies in some finite-dimensional subextension KE.
  2. 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.)
  3. Prove that ¯Q is an algebraic closure of Q. (Hint: to show that ¯Q is algebraically closed, suppose we have an algebraic extension ¯QE and choose αE. 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 QL 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.)
  4. 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.
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Questions:[edit]

Solutions:[edit]

Definitions:[edit]

  1. 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

    32 is not constructible.

Theorems:[edit]

  1. 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=E0E1En

    with αEn and, for each i between 1 and n, [Ei:Ei1]=2.