I am wondering if the splitting field of a torus is the splitting field of the polynomials defining the torus. Is for this reason that the name "splitting field" is used in both cases? Or the two fields do not coincide in general?
2026-04-03 11:42:34.1775216554
Splitting field of a torus is the splitting fields of the polynomials defining the torus?
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First of all, there's no such thing as "the" polynomials defining a torus, or any affine variety more generally. Choosing a set of polynomials to define an affine variety requires both choosing an embedding of that variety into some affine space $\mathbb{A}^n$ and then choosing some set of generators of the ideal of polynomials vanishing on the image; both of these choices are highly non-unique.
Second of all, there's no such thing as "the splitting field" of a polynomial in more than one variable; in more than one variable polynomials can remain irreducible even over an algebraically closed field.
There is a general notion of "splitting field" which contains both of these as special cases but it would take awhile to explain; it involves Galois descent.