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The addition operation can be seen to come from the fact that the extension forms a vector space over the base field. But the multiplication operation does not seem to be that obvious. Normally people just memorize the technique for multiplying two complex numbers.

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    $\begingroup$ What do you mean by "field extension"? In many cases I would think the field on top already has a multiplication by definition of being a field. $\endgroup$ – MCT Feb 24 '18 at 16:54
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It is common to construct the extension field as a quotient of the ring of polynomials $\mathbb{F}[x]$ modulo an ideal. So the multiplication operation is polynomial multiplication (followed by cancelling out stuff in the ideal).

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