Math 361, Spring 2017, Assignment 11

Carefully define the following terms, then give one example and one non-example of each:

1. Compass-and-straightedge construction.
2. Constructible number.

Carefully state the following theorems (you do not need to prove them):

1. Theorem concerning sums, differences, products, and inverses of constructible numbers.
2. Theorem concerning square roots of constructible numbers.
3. Theorem concerning the degrees of constructible numbers over $\mathbb{Q}$.
4. Theorem concerning duplication of the cube.
5. Theorem concerning squaring the circle.

Describe the following compass-and-straightedge constructions:

1. Perpendicular bisector.
2. Dropping a perpendicular.
3. Constructing a parallel.
4. Multiplication of lengths.
5. Inversion of length.
6. Square root of length.

Solve the following problems:

1. Determine whether the following numbers are constructible:

(a) $\frac{\sqrt{5}-1}{4}$.

(b) $\sqrt[6]{7}$.

(c) $\alpha$ where $\alpha$ is a root of the polynomial $x^3+3x-12$. (Hint: use Eisenstein's criterion.)

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