Example
It is not, for example, immediately obvious that if one adjoins to the field Q of rational numbers roots of both polynomials
and
say α and β respectively, to get a field K = Q(α, β) of degree 4 over Q, that the extension is simple and there exists a primitive element γ in K so that K = Q(γ). One can in fact check that with
the powers γ i for 0 ≤ i ≤ 3 can be written out as linear combinations of 1, α, β and αβ with integer coefficients. Taking these as a system of linear equations, or by factoring, one can solve for α and β over Q(γ) (one gets, for instance, α=), which implies that this choice of γ is indeed a primitive element in this example. A simpler argument, assuming the knowledge of all the subfields as given by Galois theory, is to note the independence of 1, α, β and αβ over the rationals; this shows that the subfield generated by γ cannot be that generated α or β, nor in fact that generated by αβ, exhausting all the subfields of degree 2. Therefore it must be the whole field.
Read more about this topic: Primitive Element Theorem
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“Our intellect is not the most subtle, the most powerful, the most appropriate, instrument for revealing the truth. It is life that, little by little, example by example, permits us to see that what is most important to our heart, or to our mind, is learned not by reasoning but through other agencies. Then it is that the intellect, observing their superiority, abdicates its control to them upon reasoned grounds and agrees to become their collaborator and lackey.”
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