0
$\begingroup$
  1. F, K, L are fields. F is extended to K, and K is extended to L. Show that $[L:F]=[L:K] \cdot [K:F]$.

  2. Also consider the extension from $\mathbb{Q}$ to $\mathbb{Q}(\alpha)$ where $\alpha = \sqrt{3} + \sqrt{5}$. Find the degree of this extension and the minimal polynomial of $\alpha$ over $F$.

  3. Show that any finite field extension is algebraic.

Note: This was the last assignment and my teacher decided not to post answers since she's preparing the final exam. Thank you for the help.

$\endgroup$
2
$\begingroup$

These might be better posted as three separate questions, but I'll try to answer them together.

  1. This is actually a very common proof in most algebra textbooks, so I'm surprised it's been left as an exercise. Neverthless, a hint: consider what a basis of $F$ as a vector space over $L$ looks like. If you can't progress from here, http://en.wikipedia.org/wiki/Degree_of_a_field_extension might help.

  2. The degree of the field extension is equal to the degree of the minimal polynomial $m_{\alpha, F}$ of $\alpha$ over $F$, so it suffices to find $m_{\alpha, F}(x)$. A way you might proceed would be to look at powers of $\alpha$ and see when you can "stop", i.e., build a linearly dependent relation among powers of $\alpha$; if you need more help from here, please post and I can provide more.

  3. Hint: Since the field extension is finite, then for any element $\alpha$ of the field extension, there exists a linearly dependent relation among powers of $\alpha$. Again, if you can't progress from here, http://www.proofwiki.org/wiki/Finite_Field_Extension_is_Algebraic can help.

| cite | improve this answer | |
$\endgroup$
0
$\begingroup$

What is the degree of $\mathbb C$ over $\mathbb R$? Another approach would be, what is the minimal polynomial of $\mathbb C$ over $\mathbb R$?

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.