QuadraticExtension

QuadraticExtension - Math 461 F Spring 2011 Quadratic Field...

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Unformatted text preview: Math 461 F Spring 2011 Quadratic Field Extensions Drew Armstrong Let F be a field and let c F be an element such that c 6 F . (This notation means that the equation x 2- c = 0 has no solution in F .) In this case we can define a new, bigger number system F [ c ] := { a + b c : a,b F } , which we call F adjoin c . We have already seen an important example of this. The complex numbers are just the same as R adjoin - 1: C = R [ - 1] = a + b - 1 : a,b R . You will agree by now that the complex numbers have remarkable and beau- tiful properties. So perhaps the same is true of F [ c ]? Yes. First note that we can divide in F [ c ]. Given a + b c F [ c ] we have 1 a + b c = 1 a + b c a- b c a- b c = a- b c a 2- cb 2 = a a 2- cb 2 +- b a 2- cb 2 c, which is again in F [ c ]. We can multiply, add, and subtract elements of F [ c ] in the obvious way. Hence F [ c ] is itself a field . We will call the pair....
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QuadraticExtension - Math 461 F Spring 2011 Quadratic Field...

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