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.


2 Answers 2


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.


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


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