Primitive elements

Dear Dr Kim,

There is a post on your blog regarding finding primitive elements. Your advice was to look at the Primitive elements: an example document. Can I assume this would be an acceptable answer in the exam and, would just stating that the method is the same as the one used for proving the Primitive Element Theorem be sufficient justification or do we need to provide further explanation?

Thanks in advance,

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Reply:

Of course the method is acceptable, but I don’t understand what you mean by `sufficient justification.’

Let’s remind ourselves what method we are speaking of:

To find a primitive element in Q(\alpha, \beta), we need to locate a linear combination \alpha + c\beta with c\in Q satisfying certain conditions spelled out in the proof of the Primitive Element Theorem. It might be \alpha+\beta, \alpha-\beta, \alpha+(1/2)\beta, etc. depending on the situation, even though the result tends to be rather simple in the examples that have come up. To use the method of the theorem would mean checking that the conditions are satified for some specific c. If you did this, yes it would be sufficient justification.

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