Homework Assignment 1

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 7.3: 19, 25

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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 7.3: 3, 6, 10, 11, 16, 17, 18, 26, 30

1. Additional Problem. Suppose $R$ is a ring and $c\in R$. Let $I = \{p(x) \in R[x] \mid p(c) = 0\}$.
   1. Prove that $I$ is an ideal of $R[x]$.
   2. Use the First Isomorphism Theorem to prove that $R[x]/I \cong R$ as rings.

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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 7.3: 24

2. Section 7.3: 29

   The Binomial Theorem is given in Problem #25 of Section 7.3, and you are welcome to use it. You’re also welcome to use Problem #14 of Section 7.1, which you proved last semester.