Tensor products and base-field extensions

Up to our discussion of eigenvalues and eigenvectors in section 9 the base-field of a vector space did not play any role whatsoever. Only the basic field axioms entered into the discussion, but no special properties of the field. ^† However, when we want to find the eigenvalues of an endomorphism A: V → V of a finite-dimensional K-vector space V, we have to find the roots of the characteristic polynomial p(λ) = det(A − λ1) ∈ K[λ], and the number of roots of a polynomial depends on the base-field. For example, the polynomial λ² − λ − 1 has no root in ℚ, but has two roots in ℚ + ℚ 5 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315136547/a57ae1f0-6f59-46bd-bbdd-35a521b486a1/content/eq1207.tif"/> ; similarly, the polynomial λ³ + λ only has the root 0 in ℝ but has three roots in ℂ. It was pointed out already (and will be proved in the field theory chapter in Volume II of this book) that every field K can be extended to a field K ¯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315136547/a57ae1f0-6f59-46bd-bbdd-35a521b486a1/content/eq1208.tif"/> such that every polynomial f ∈ K [λ] of degree n has n roots in K ¯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315136547/a57ae1f0-6f59-46bd-bbdd-35a521b486a1/content/eq1209.tif"/> . Thus for a given endomorphism A: V → V we can always find n:= dim V roots of the characteristic polynomial of A. But can we still meaningfully interpret these roots as “eigenvalues” of A if they do not lie in the original base-field K? To do so, it is necessary to find along with the field extension K ⊂ K ¯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315136547/a57ae1f0-6f59-46bd-bbdd-35a521b486a1/content/eq1210.tif"/> an extension of V to a vector space V ¯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315136547/a57ae1f0-6f59-46bd-bbdd-35a521b486a1/content/eq1211.tif"/> over K ¯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315136547/a57ae1f0-6f59-46bd-bbdd-35a521b486a1/content/eq1212.tif"/> in which the corresponding eigenvectors can be found.