Homework Assignment 3

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 7.5: 3, 4
• Section 8.1: 11
• Section 8.2: 3, 7(a)
• Section 8.3: 6(a,b)
• Section 9.1: 5, 6

1. Additional Problem. Suppose $R$ is a PID. Quickly prove (by connecting a couple of propositions) that a nonzero irreducible element generates a maximal ideal.

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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 8.1: 3

2. Section 8.1: 10

   Follow the hint in the book using the norm $N(a+bi) = a^2 + b^2$. As an intermediate step, explain why for each $m\in \mathbb{N}$, there are only finitely many elements of $\mathbb{Z}[i]$ of norm $m$.

3. Section 9.1: 4