Let L and N be field extension of K.

Tensor product of L and N over K is always field？

If there are counterexamples, I would be appreciated if you could give me an example and explanation.

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# Tensor product of fields may not be a field

###### Related Question

field-theorytensor-products

Let L and N be field extension of K.

Tensor product of L and N over K is always field？

If there are counterexamples, I would be appreciated if you could give me an example and explanation.

## Best Answer

Given any proper field extension $K \subset L$, the tensor product $L \otimes_K L$ is never a field, because the kernel of the multiplication map $L \otimes_K L \to L$ is a nonzero proper ideal.