08-401/Homework Assignment 8

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Reading

Read chapter 32 of Gallian's book (6th edition) 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.

Also, spend some time reviewing what you've learned in the past about group theory, with special emphasis on normal subgroups, quotients and the isomorphism theorems.

Doing

Solve problems 1, S2 ($GF(4):=({\mathbb {Z} }/2)[x]/\langle x^{2}+x+1\rangle$ is the field with 4 elements), S4, S6, 7, S8, 9, S14 ($D_{5}$ is defined on page 34), S15, 16, 17, and 18 in chapter 32 of Gallian's book (6th edition), but submit only the solutions of the problems marked with the letter "S".

Due Date

This assignment is due in class on Wednesday March 26, 2008.

Just for Fun

Let $x$ be some root of the equation

$x^{5}+({\sqrt[{3}]{2}}-{\sqrt {3}})x^{4}+{\frac {1}{\sqrt {{\sqrt {3}}+{\sqrt {5}}}}}x^{3}-1=0.$

We know that $x$ is algebraic. Can you find a polynomial $f$ with rational coefficients whose roots include $x$? Can you find the minimal polynomial $p$ of $x$? 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 such a polynomial $f$ through your own efforts, post it on this site along with your detailed and repeatable computations leading to it (a computer program, I presume, fully documented so that anybody else could run it too), and you will receive up to 60 bonus points for the feat, counted towards your final grade in the same manner as the "good deed" points. The due date for prize claims is the last day of classes.