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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?

lmonninger
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    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. – Greg Martin May 14 '22 at 05:15
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    Something like this is discussed at https://physics.stackexchange.com/questions/52176/could-negative-dimension-ever-make-sense See also https://math.stackexchange.com/questions/423874/do-negative-dimensions-make-sense and the questions linked there. – Gerry Myerson May 14 '22 at 06:56
  • @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? – lmonninger May 14 '22 at 17:14
  • @GregMartin Separately, for $L()$ does the description of a field extension degree as the degree of the minimal polynomial over $L$ break down because of the general resistance to consider polynomials of negative degrees? What if we accept Laurent polynomials? – lmonninger May 14 '22 at 17:21
  • @GerryMyerson The content you've linked to, together with Greg Martin's comment, is making me lean towards conceptualizing this inquiry as fundamentally concerning the constraints placed on counting. Does this seem like a valid a takeaway? – lmonninger May 14 '22 at 17:25
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    If it's constraints on counting that interest you, you might have a look at https://math.stackexchange.com/questions/73470/fractional-cardinalities-of-sets/ and the links given there to work of John Baez. – Gerry Myerson May 15 '22 at 00:35
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    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. – Gerry Myerson May 15 '22 at 00:39

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