07-401/Homework Assignment 7

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Reading

Read chapters 20 and 21 of Gallian's book three times:

• First time as if you were reading a novel - quickly and without too much attention to detail, just to learn what the main keywords and concepts and goals are.
• Second time like you were studying for an exam on the subject - slowly and not skipping anything, verifying every little detail.
• And then a third time, again at a quicker pace, to remind yourself of the bigger picture all those little details are there to paint.

Doing

Solve problems 20 and 27# in Chapter 20 of Gallian's book and problems 3#, 7, 8#, 9, 10# and 18# in Chapter 21 of the same book, but submit only the solutions the problems marked with a sharp (#).

Due Date

This assignment is due in class on Wednesday March 21, 2007.

Just for Fun

We know that if $a$ and $b$ are algebraic numbers, then so are $a+b$, $a-b$, $ab$, $a/b$ and ${\sqrt[{n}]{a}}$ (for any natural $n$). It follows that the number

$c={\sqrt[{3}]{7}}-{\sqrt[{4}]{{\sqrt {5}}\left/{\sqrt[{3}]{{\sqrt {2}}+{\sqrt[{5}]{7}}}}\right.}}$

is algebraic. If so, can you find a polynomial whose roots include $c$? Can you find the minimal polynomial $p$ of $c$? What is $\deg p$?

Warning. "Can you find?" should be interpreted as "How would you find?" and not as "Please find.". The latter is doable, but not by hand!

Prize. Though if you do actually find the polynomial $p$ through your own efforts, post it on this site along with your computations leading to it (a computer program, I presume), and your final grade for this class will be bounded below by 90. The due date for prize claims is the last day of classes.

Solutions 1. Andrei Litvin (partial, however should be quite close)