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Problem 65 (Every Element is a Product of Irreducibles in a PID)
Suppose that $D$ is a principle ideal domain with $a\in D$ where $a\neq 0$ is not a unit. Prove that $a$ can be written as the product of irreducibles.
Hint: You'll want to use the previous problem quite a bit. If $a$ is irreducible, then you're done. You'll probably want to start by showing that you can write $a=pc$ for some irreducible $p$ (in other words, you can factor off an irreducible. How will you use the previous problem? If you can't factor off an irreducible, then you can write $a=a_1b_1$ for two non unit non irreducible terms. Why do we know $\langle a\rangle\subseteq\langle a_1\rangle$? Now repeat this process to get $\langle a\rangle\subseteq\langle a_1\rangle\subseteq\langle a_2\rangle$ and so on. Once you've gotten $a=pc$ for some irreducible $p$, what connection is there between $\langle a\rangle$ and $\langle c\rangle$?
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