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I've only ever seen finite field extensions indicated as $[L:K] < \infty$. I've never seen $-\infty<[L:K]<\infty$. I take this to mean that field extensions of a negative degrees are not considered.

This makes sense, because I have no idea how you would get a field extension of a negative degree. However, I'm curious, is there an interpretation of field theory that would allow for this? If so, what would a negative degree mean?

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    $\begingroup$ I doubt there's any reasonable interpretation like this. The degree of a field extension literally counts something (namely the number of elements in a basis of $L$ considered as a vector space over $K$), which is why it's a positive integer. $\endgroup$
    – Greg Martin
    May 14 at 5:15
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    $\begingroup$ Something like this is discussed at physics.stackexchange.com/questions/52176/… See also math.stackexchange.com/questions/423874/… and the questions linked there. $\endgroup$
    – Gerry Myerson
    May 14 at 6:56
  • $\begingroup$ @GregMartin Would a metric $[L : K]'$ representing the number of elements in a basis $L$ not considered as a vector space over $K$ be reasonable? $\endgroup$
    – lmonninger
    May 14 at 17:14
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    $\begingroup$ If it's constraints on counting that interest you, you might have a look at math.stackexchange.com/questions/73470/… and the links given there to work of John Baez. $\endgroup$
    – Gerry Myerson
    2 days ago
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    $\begingroup$ Let me note that given any Laurent polynomial, there is an ordinary polynomial giving rise to the same field extension, so as far as extension fields are concerned, you gain nothing by allowing Laurent polynomials. $\endgroup$
    – Gerry Myerson
    2 days ago

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