1

Context: The book Galois Theory by Ian Stewart, 4th edition, Chapter 5, Exercise 5.7.

Is the field extension $\mathbb{Q} \left( \sqrt{2}, \sqrt{3}, \sqrt{5} \right)$ of the field $\mathbb{Q}$ of rational numbers a simple extension?

That is, is there an real number $\alpha$ such that $$ \mathbb{Q} ( \alpha ) = \mathbb{Q} \left( \sqrt{2}, \sqrt{3}, \sqrt{5} \right)? $$

Of course, we can show that $$ \mathbb{Q} \left( \sqrt{2} + \sqrt{3} + \sqrt{5} \right) \subset \mathbb{Q} \left( \sqrt{2}, \sqrt{3}, \sqrt{5} \right) . $$ Am I right?

Can we also show that $$ \mathbb{Q} \left( \sqrt{2}, \sqrt{3}, \sqrt{5} \right) \subset \mathbb{Q} \left( \sqrt{2} + \sqrt{3} + \sqrt{5} \right)? $$

Or, is there a way of conclusively proving that showing this is not possible?

Is there a way of showing using some other technique that this extension is or is not simple?

Saaqib Mahmood
  • 26,762
  • 1
    The primitive element theorem gives you the existence. Its constructive proof gives you a way to test that particular generator. – egorovik Dec 12 '19 at 19:51
  • Note that the minimal polynomial of $\sqrt{2}+\sqrt{3}+\sqrt{5}$ has degree 8, which is the degree of the extension field $\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5})$. – rogerl Dec 12 '19 at 20:40
  • @rogerl can you please explicitly show in a full-fledged answer that the following holds? $$\mathbb{Q}\left( \sqrt{2}, \sqrt{3}, \sqrt{5} \right) \subset \mathbb{Q} \left( \sqrt{2} + \sqrt{3} + \sqrt{5} \right). $$ – Saaqib Mahmood Dec 13 '19 at 08:53
  • @egorovik can you please explicitly show that the following holds? $$\mathbb{Q}\left( \sqrt{2}, \sqrt{3}, \sqrt{5} \right) \subset \mathbb{Q} \left( \sqrt{2} + \sqrt{3} + \sqrt{5} \right). $$ Please post your work as an answer. – Saaqib Mahmood Dec 13 '19 at 08:55
  • @SaaqibMahmood Do you have problems figuring out the answer from either my hint or egorovik's? If so, it would be helpful if you told us what confuses you. – rogerl Dec 13 '19 at 12:38
  • @rogerl actually, Ian Stewart hasn't discussed either the Primitive Element Theorem or any result involving the degree of the minimal polynomial. So I would appreciate if either you or egorovik give an explicit formulation of the required inclusion relation. – Saaqib Mahmood Dec 13 '19 at 18:18

0 Answers0