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Let $\sqrt[3]{2}$ be the real cube root of 2.

(a) Find the minimal polynomials of cube root of $2$ and $i$ over $\mathbb{Q}$.
(b) Find the minimal polynomial of $i$ over $\mathbb{Q}(\sqrt[3]{2})$.
(c) Show that $\mathbb{Q}(i\cdot\sqrt[3]{2}) = \mathbb{Q}(i,\sqrt[3]{2})$.
(d) Find the minimal polynomial of $i\cdot\sqrt[3]{2}$ over $\mathbb{Q}$.

I am lost in this setup, and this is not a homework assignment. It is just for something the teacher didn't cover due to lack of time.

For part a:

I got the answer to be x^3 -2 and x^2 + 1, respectively

For part b:

I got the answer to be x^2 + 1

part c: I am unsure what's going on, and need help. can someone explain it please?

part d: answer is x^6 + 4 = 0, right?

  • 1
    For a start: what is the minimal polynomial of cube root 2? what is the minimal polynomial of $i$?2011-05-03
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    x^3-2; i^3-2, are there correct?2011-05-03
  • 2
    The first one is correct. Why $i^3-2$?2011-05-03
  • 2
    I posted something about c) ten days ago. If you didn't find it helpful, maybe you could indicate what part(s) of my answer you have trouble with. This is more polite than just editing your question and ignoring (or appearing to ignore) efforts to help you.2011-05-13

2 Answers 2