Directions.
- If you want, you may work together with one partner on this and turn in the same pdf for both of you.
- If you do this, please make sure that you truly collaborate on revising the math and typing it up.
- To collaborate with Overleaf, one person should start the project, and then use the share option (on the top menu bar), to share it with their partner.
- Get the template for this assignment (or use your own). Here’s how to do it:
- Go to www.overleaf.com, and make sure you are logged in.
- In a new window, go here: https://www.overleaf.com/read/pkhszcbmsfxj
- Click on the menu icon in the upper-left and select “Copy Project”.
- When ask for a name, choose something like “Math 110B - WA 07” and click “Copy”.
- When this completes you will be back in your own workspace (instead of mine).
- Use $\LaTeX$ to type up your proofs of the Propositions to Prove listed below. Make sure to use complete sentences and appropriate punctuation. Also, make sure to edit for typos.
- Click on the “Download PDF” button (to the right of the “Recompile” button). Save the file somewhere you can easily find it. Submit the pdf in Canvas.
- Let me know if you have any questions!
Remember, if you have trouble finding the command for a math symbol you want to use, try googling it, email me, or try looking in this Short Math Guide for $\LaTeX{}$.
Propositions to Prove.
- Let $\phi$ be the “evaluation at $\sqrt{5}$” homomorphism $\phi:\mathbb{Q}[x]\rightarrow\mathbb{C}$ defined by $\phi(p(x)) = p(\sqrt{5})$. Then $\ker \phi = (x^2 -5)$, and $\operatorname{im}\phi = \mathbb{Q}(\sqrt{5})$. Consequently, $\mathbb{Q}[x]/(x^2-5) \cong \mathbb{Q}(\sqrt{5})$.
You’ll need to adapt the ideas from Problem 5.118 and 5.125 to work with $\sqrt{5}$ instead of $i$. Remember that $(x^2 -5)$ denotes the principal ideal of $\mathbb{Q}[x]$ generated by $x^2 -5$. Make sure to prove all three statements: $\ker \phi = (x^2 -5)$, $\operatorname{im}\phi = \mathbb{Q}(\sqrt{5})$, and $\mathbb{Q}[x]/(x^2-5) \cong \mathbb{Q}(\sqrt{5})$.
