## 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 $\measuredangle 36^\circ.$]

### 5 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. andrescaicedo says:

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!

5. […] Homework 6, due April 3, at the beginning of lecture. […]