Homework Assignment 4

Results to know

Please read over these problems. You can use these results in future problems even though you are not being asked to prove them.

• Section 9.4: 17

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Problems to try

Please try to solve these problems, but you do not turn them in. You can also use these results in future problems. I’m happy to talk about these problems (or any others) if you have questions.

• Section 9.2: 1, 2, 3, 6
  • On #1, the parenthetical statement is just for context and is automatically true once you prove the main statement.
• Section 9.4: 1, 3, 16
  • On #1(c), it may be helpful to note that $1\equiv -4$ (mod 5).

1. Additional Problem. Let $F$ be a field, and let $p(x) \in F[x]$. For all $a,b\in F$ with $a\neq 0$, if $p(x)$ is reducible in $F[x]$, then $p(ax+b)$ is also reducible in $F[x]$.
   • We typically use this in the contrapostive form: if $p(ax+b)$ is irreducible, then so is $p(x)$. This also holds with a higher degree polynomial replacing $ax+b$.

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Problems to submit

These are the problems to turn in for a grade. Please write in complete sentences, use correct punctuation, and organize your work. Once your finish these problems, please follow the directions in the Canvas assignment to submit them. If you have any questions (about the math or writing or submission process or anything), please let me know!

1. Section 9.4: 2

2. Section 9.4: 6

   Your answer to each part should be something like “Consider the field $K = F[x]/(p(x))$ where $F = $ (fill in the blank) and $p(x) = $ (fill in the blank). Then $K$ is a field of order (fill in the blank) because… (insert justification including proof that $p(x)$ is irreducible)”

3. Section 9.5: 7