## Primitive elements

Dr Kim,

Concerning Primitive Elements, i have a few quieries about calculations. In the 2008 past paper, we have the example of finding the primitive element of

$Q(\sqrt{3},\sqrt{5}).$

The solutions begin with assuming this is $\sqrt{3}+\sqrt{5}$. Can we always do this? If not what method do we use to find it? Then we find the degree (of the field generated by the primitive element?) has to be 4 by the tower theorem. I understand this, but in another example if the degree was not the same as the field extension (ie.not 4) what do we do then/how do we define the subfield etc.?

Many thanks,
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