I am reading Anderson and Feil - A First Course in Abstract Algebra.(adsbygoogle = window.adsbygoogle || []).push({});

I am currently focused on Ch. 44: Finite Extensions and Constructibility Revisited ... ...

I need some help in fully understanding Example 44.2 ... ...

Example 44.2 reads as follows:

I am trying to fully understandwhyEXACTLY

##\{ 1, \sqrt{2}, \sqrt{3}, \sqrt{6} \}##

is the basis chosen for ##\mathbb{Q} ( \sqrt{2}, \sqrt{3} )## ... ...

I can see why ##1, \sqrt{2}, \sqrt{3}## are in the basis ... and I understand that we need ##4## elements in the basis ....

... BUT ... whydo we add ##\sqrt{6} = \sqrt{2} \cdot \sqrt{3}## ... an element that is already in the set generated by ##1, \sqrt{2}, \sqrt{3}## ...EXACTLY

... indeed, what is the rigorous justification for adding ##\sqrt{6}## ... why not add some other element ... ... for example, why not add ##\sqrt{12}## ...

Hope someone can help ...

Peter

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# I Finite Extensions - A&F Example 44.2 ... ...

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