2

Let $\xi$ be the the fifth root of unity $\ne 1$, and let $z=\xi 2^{1/5}$ and $t=z^2+z^3$

I have to find an explicit expression for the minimal polynomial of $t$ over $\Bbb{Q}$.

I am stuck, I can prove that the splitting field of $X^5-2$ over $\Bbb{Q}$ is $\Bbb{Q}(\xi, 2^1/5)$ so the degree is $20$.

I can prove that $\Bbb{Q}(t)$ is included in $\Bbb{Q}(z)$. But then I am not sure what I have to "do".

JeSuis
  • 697

1 Answers1

4

You may simply examine the field $ \mathbf Q(z) \cong \mathbf Q(2^{1/5}) $. Using the obvious basis and writing out the multiplication map by $ 2^{2/5} + 2^{3/5} $ in matrix form gives

$$ \begin{bmatrix} 0 & 0 & 2 & 2 & 0 \\ 0 & 0 & 0 & 2 & 2 \\ 1 & 0 & 0 & 0 & 2 \\ 1 & 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 \end{bmatrix} $$

Now, compute the characteristic polynomial of this matrix. Since $ 2^{2/5} + 2^{3/5} $ is irrational, it is a primitive element, thus the characteristic polynomial will be equal to its minimal polynomial.

Ege Erdil
  • 17,747
  • For any field extension $ L/K $, multiplication by an element of $ L $ induces a $ K $-linear map, which can be written in matrix form by fixing a basis of $ L/K $ - this is just linear algebra. – Ege Erdil Oct 06 '16 at 12:58
  • Well, you know what the roots are - they are the five distinct $ \mathbf Q $-conjugates of $ 2^{2/5} + 2^{3/5} $. – Ege Erdil Oct 10 '16 at 15:43
  • If an extension of fields $ L/K $ has prime degree, then since the degree of an element over $ K $ divides the degree of the extension, it follows that an element is either in $ K $, or generates $ L $ over $ K $. The $ \mathbf Q $-conjugates of an algebraic number are the roots of its minimal polynomial over $ \mathbf Q $. – Ege Erdil Oct 10 '16 at 15:54
  • @JeSuis: I am no sure that is a kind or acceptable approach. Starfall's answer is fine, so, what are you going to do, reward the bounty to an answer you previously un-accepted? – Jack D'Aurizio Oct 11 '16 at 11:07
  • 1
    @JeSuis: all right, but in such a case you should state in your question what instruments are allowed / at your disposal to solve the given problem. Starfall's answer is fine and should not be "punished" just because someone thinks it is "too advanced", also because it is not. The concepts of conjugates and primitive elements are central in field theory, and nothing restrains you from acquiring this knowledge through a quick search on Wikipedia, if your textbook does not mention them (kind of strange). – Jack D'Aurizio Oct 11 '16 at 11:25
  • This similar question should enlighten you, too: http://math.stackexchange.com/questions/1204279/show-that-x4-10x21-is-irreducible-over-mathbbq – Jack D'Aurizio Oct 11 '16 at 11:27