07-401/Homework Assignment 7

2007-03-26T00:04:05Z

76.64.136.153:

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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 <math>a</math> and <math>b</math> are algebraic numbers, then so are <math>a+b</math>, <math>a-b</math>, <math>ab</math>, <math>a/b</math> and <math>\sqrt[n]{a}</math> (for any natural <math>n</math>). It follows that the number
{{Equation*|<math>c=\sqrt[3]{7}-\sqrt[4]{\sqrt{5}\left/\sqrt[3]{\sqrt{2}+\sqrt[5]{7}}\right.}</math>}}
is algebraic. If so, can you find a polynomial whose roots include <math>c</math>? Can you find the minimal polynomial <math>p</math> of <math>c</math>? What is <math>\deg p</math>?

'''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 <math>p</math> ''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. [[07-401/Andrei Litvin Poly Solution|Andrei Litvin]] (partial, however should be quite close)

07-401/Andrei Litvin Poly Solution

2007-03-26T00:03:59Z

76.64.136.153:

Here is what I have so far in terms of polynomials:

Polynomial with root <math>\sqrt{2}</math>:

<math>{x}^{2} - 2</math> (duh)

Polynomial with root <math>\sqrt[5]{7}</math>:

<math>x^5 - 7</math> (duh as well)

Polynomial with root <math>\sqrt{2} + \sqrt[5]{7}</math>:

<math>x^{10} - 10*x^8 + 40*x^6 - 14*x^5 - 80*x^4 - 280*x^3 + 80*x^2 - 280*x + 17
</math> (more interesting!)

Polynomial with root <math>\sqrt[3]{\sqrt{2} + \sqrt[5]{7}}</math>:

<math>x^{30} - 10*x^{24} + 40*x^{18} - 14*x^{15} - 80*x^{12} - 280*x^9 + 80*x^6 - 280*x^3 + 17
</math> (similar to the above)

Polynomial with root <math> {1} \over {\sqrt[3]{\sqrt{2} + \sqrt[5]{7}}}</math>:

<math>
17*x^{30} - 280*x^{27} + 80*x^{24} - 280*x^{21} - 80*x^{18} - 14*x^{15} + 40*x^{12} - 10*x^6 + 1
</math> (just invert the above .. quite the same as the above 2)

Polynomial with root <math> {\sqrt{5}} \over {\sqrt[3]{\sqrt{2} + \sqrt[5]{7}}}</math>:

<math>
289*x^{60} - 9460000*x^{54} - 2392500000*x^{48} - 190781250000*x^{42} + 1127929687500*x^{36} - 249084472656250*x^{30} + 12817382812500000*x^{24} - 457763671875000000*x^{18} + 10728836059570312500*x^{12} - 149011611938476562500*x^6 + 931322574615478515625
</math> (ok .. this is ugly)

Polynomial with root <math> \sqrt[4]{{\sqrt{5}} \over {\sqrt[3]{\sqrt{2} + \sqrt[5]{7}}}}</math>:

<math>
289*x^{240} - 9460000*x^{216} - 2392500000*x^{192} - 190781250000*x^{168} +
1127929687500*x^{144} - 249084472656250*x^{120} + 12817382812500000*x^{96} -
457763671875000000*x^{72} + 10728836059570312500*x^{48} - 149011611938476562500*x^{24} + 931322574615478515625
</math> (very ugly .. especially considering you are not done yet and the next matrix to compute will be quite large)

This is how far I got so far. The program itself will do others just file (I fiddled with it a lot, so it may still have some bugs, however it works). The problem is working with such large numbers. The reducing part will multiply with 289 each time a <math>x^{240}</math> is reduced, so we could potentially get coefficients of the order <math>289^{964 - 240}</math>, which is very large indeed. On my computer, I get about 30% done, when it starts running out of memory and I had to stop. This is why the program just generates the matrices for the last part (and by the way, Maxima crashed when trying to load the matrix to invert, so I will have to find another program that will handle such large sets of data)

Drorbn - User contributions [en]

07-401/Homework Assignment 7

07-401/Andrei Litvin Poly Solution