Classify the fields such that the ring is again a field.

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# Tensor product of fields

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3 thoughts on “Tensor product of fields”

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Classify the fields such that the ring is again a field.

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HINT: consider the multiplication map .

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I claim this is true if and only if is a prime field, that is, is either the rationals or a finite field of order , a prime. Indeed it is easy to see that is isomorphic to where is the prime subfield of . If this is a field, then multiplication (as in Luis’ comment) is injective, from where and ; whence .

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There is a closely related formula by Grothendieck:

Let be field extensions. Then .

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