Basic Concepts

Let K / k https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003177036/f0af01d0-3a24-4c34-9b44-c23e4a0fe932/content/math3_1.tif"/> be a field extension and suppose α is an element of K. We say that α is algebraic overk if α satisfies a nonzero polynomial over k. Suppose n = dim ⁡ k ( K ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003177036/f0af01d0-3a24-4c34-9b44-c23e4a0fe932/content/math3_2.tif"/> is finite and α is in K. Then the n + 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003177036/f0af01d0-3a24-4c34-9b44-c23e4a0fe932/content/math3_3.tif"/> vectors 1 , α , … , α n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003177036/f0af01d0-3a24-4c34-9b44-c23e4a0fe932/content/math3_4.tif"/> cannot be linearly independent and hence satisfy a nontrivial linear relation c 0 + c 1 α + ⋯ + c n α n = 0 , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003177036/f0af01d0-3a24-4c34-9b44-c23e4a0fe932/content/math3_5.tif"/>