## 305 -Homework set 6

This set is due April 3 at the beginning of lecture. Details of the homework policy can be found on the syllabus and here.

1. Find ${\mathbb Q}^{p(x)}$ where $p(x)=x^3-2,$ and determine all its subfields. Make sure you justify your answer. For example, if you state that two subfields ${\mathbb F}_1$ and ${\mathbb F}_2$ are different, you need to prove that this is indeed the case.

2. Do the same for $p(x)=x^4+x^3+x^2+x+1.$

[Updated, April 2: I guess the hint I gave for problem 2 makes no sense, sorry about that. Rather, you may want to begin by looking at how $x^5-1$ factors. Then, to compute $\cos(72^\circ),$ it may be helpful to look at a triangle with angles $\measuredangle 72^\circ,$ $\measuredangle 72^\circ,$ and $\measuredangle 36^\circ.$]

### 4 Responses to 305 -Homework set 6

1. […] Homework 6, due April 3, at the beginning of lecture. Possibly related posts: (automatically generated)117b – Homework 5Buffalostyle Forges OnHomework battles and the biggest genius in the school, part IThe Myth About Homework […]

2. Tommy says:

Concerning HW assignment 6, I am wondering if there is a typo in the hint you gave us for problem number 2. I may be wrong but I believe the denominator under the radical on the left hand side should be 18. Thanks!

3. Hi Tommy,

Hmm, yeah. I’m just about convinced now that the hint is nonsense. So, I have added another hint.

4. Karen says:

Hi,
Okay…I am struggling with the second problem. I have solved the quartic down to w^6 and found my value for w. Then, when I plug everything back in, I cannot get any given u to solve the equation where u^3 + ….=0. Therefore I am second guessing everything I have done. And I appreciate the new hint, but I’m not sure how to apply it. I have worked the equation many times, each time hoping my calculations are wrong…unfortunately so far, they are not.
Any help would be greatly appreciated!