Over $\mathbb R^n$, the standard intuition given to the determinant is that it measures the signed area of the image of an unit cube. But determinants can be more generally defined for endomorphisms of spaces of finite dimension over other fields than $\mathbb R$, where this notion of area doesn't exist, or even over other rings, or semirings or other things. Is there more general intuition for what the determinant means that isn't specific to the field $\mathbb R$?
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