Tuesday, November 6, 2018

Exercise about field extensions




Consider $a_1,\ldots,a_n\in \mathbb Z$.



i) Suppose $a_1,\ldots, a_n$ are pairwise relatively prime. I have to see by induction on n that $[\mathbb Q(\sqrt a_1,\ldots,\sqrt a_n):\mathbb Q]=2^n$



Once I proved the equality is true for $n=1$, I suppose it is true for $n-1$, so let's prove it for $n$:
Applying the tower law:
$[\mathbb Q(\sqrt a_1,\ldots,\sqrt a_n):\mathbb Q]=[\mathbb Q(\sqrt a_1,\ldots,\sqrt a_n):\mathbb Q(\sqrt a_1,\ldots,\sqrt a_{n-1})] [\mathbb Q(\sqrt a_1,\ldots,\sqrt a_{n-1}):\mathbb Q]$



By induction, $[\mathbb Q(\sqrt a_1,\ldots,\sqrt a_{n-1}):\mathbb Q]=2^{n-1}$




So we only have to see that $[\mathbb Q(\sqrt a_1,\ldots,\sqrt a_n):\mathbb Q(\sqrt a_1,...,\sqrt a_{n-1})]=2$



$[\mathbb Q(\sqrt a_1,\ldots,\sqrt a_n):\mathbb Q(\sqrt a_1,\ldots,\sqrt a_{n-1})]=[\mathbb Q(\sqrt a_1,\ldots,\sqrt a_{n-1})(a_n):\mathbb Q(\sqrt a_1,\ldots,\sqrt a_{n-1})]=deg(Irr(\sqrt a_n,\mathbb Q(\sqrt a_1,\ldots,\sqrt a_n))$



$Irr(\sqrt a_n,\mathbb Q(\sqrt a_1,\ldots,\sqrt a_{n-1})=X^2-a_n??$



I have to see that $\sqrt a_n \notin Q(\sqrt a_1\ldots,\sqrt a_{n-1})$. By contradiction,



if$ \sqrt a_n \in Q(\sqrt a_1,\ldots,\sqrt a_{n-1}) \to \sqrt a_n= a+b\sqrt a_{n-1}$ where $a,b\in Q(\sqrt a_1,\ldots,\sqrt a_{n-1}$. How can I get to a contradiction???




ii) Consider $P$ the set of prime numbers and $F$ an extension of $\mathbb Q$: $F=\mathbb Q (\sqrt p, p\in P)$ Which is the degree of $F/\mathbb Q$? Is it finitely generated?



Could you help me with this problem please?



Thank you for your time and help.


Answer



I think that the most elegant proof of this question, which is a consequence of the fact that



Square roots of different square free positive integers are linearly independent over $\mathbb Q$,




can be found in:



http://www.thehcmr.org/issue2_1/mfp.pdf


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