This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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....
View Full
Document
 Spring '11
 Armstrong
 Math, Algebra

Click to edit the document details