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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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This note was uploaded on 01/08/2012 for the course MATH 561 taught by Professor Armstrong during the Spring '11 term at University of Miami.
 Spring '11
 Armstrong
 Math, Algebra

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