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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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