Math 380, Spring 2018, Assignment 4

I was at the mathematical school, where the master taught his pupils after a method scarce imaginable to us in Europe. The proposition and demonstration were fairly written on a thin wafer, with ink composed of a cephalic tincture. This the student was to swallow upon a fasting stomach, and for three days following eat nothing but bread and water. As the wafer digested the tincture mounted to the brain, bearing the proposition along with it.

- Jonathan Swift, Gulliver's Travels

Read:

1. Section 1.5.
2. Section 2.1.

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

1. Monic polynomial (in $\mathsf{k}[x]$).
2. $\mathrm{gcd}(f,g)$ (where $f,g\in\mathsf{k}[x]$).

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

1. Theorem concerning the division algorithm.
2. Classification of ideals in $\mathsf{k}[x]$ ("Every ideal in $\mathsf{k}[x]$ is generated by...").
3. Theorem concerning monic generators of ideals in $\mathsf{k}[x]$.
4. Theorem relating $\mathrm{gcd}(f,g)$ to common divisors of $f$ and $g$.
5. Theorem relating $\left\langle f,g\right\rangle$ to $\left\langle g,r\right\rangle$ when $f=gq+r$.

Carefully describe the following algorithms:

1. Division algorithm (to compute quotient and remainder when $f$ is divided by $g$).
2. Euclid's algorithm (to compute $\mathrm{gcd}(f,g)$).
3. Algorithm to replace any system of finitely many univariate equations by a single univariate equation.
4. Algorithm to factor polynomials over $\mathbb{C}$. (Note: this algorithm is conceptually simple but should NOT be used in engineering applications; it is numerically unstable. See books on numerical analysis for more practical algorithms.)

Solve the following problems:

1. Section 1.5, problems 1, 3, 4, 11, and 12.
2. Prove that for any $f_1,\dots,f_s\in\mathsf{k}[x]$, one has $\left\langle f_1,f_2,f_3,\dots,f_s\right\rangle = \left\langle \mathrm{gcd}(f_1,f_2),f_3,\dots,f_s\right\rangle$. (Hint: use the ideal containment criterion from the previous assignment.)
3. Optional (but interesting): section 1.5, problem 17. (You may need to use the results of problems 13, 14, and 15.)

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