Field Theory: Splitting Field
Date: 06/12/2002 at 23:31:39 From: Lucia Subject: Field theory-Abstract algebra Hello, Dr. Math. Thank you for the help you have given me in the past. Can you please help with the following: Find the splitting field of (x^3 - 5) Thank you very much. Lucia
Date: 06/13/2002 at 11:58:06 From: Doctor Paul Subject: Re: Field theory-Abstract algebra What are the three roots? Well, you know that the real root is cbrt(5). But there are two imaginary roots. Notice that if zeta_3 is a primitive third root of unity (i.e., (zeta_3)^3 = 1 and no smaller power of zeta_3 equals one) then [cbrt(5) * zeta_3]^3 - 5 = 5*1 - 5 = 0 Thus cbrt(5) * zeta_3 is also a root of x^3 - 5. Also notice that [cbrt(5) * (zeta_3)^2]^3 - 5 = 5*(zeta_3)^6 - 5 = 5*[(zeta_3)^3] ^2 - 5 = 5*1^2 - 5 = 5*1 - 5 = 0 Thus cbrt(5) * (zeta_3)^2 is also a root of x^3 - 5. There can be only three roots, so we have found them all. The splitting field is found by adjoining all of the roots to Q. Thus the splitting field is: Q(cbrt(5), cbrt(5) * zeta_3, cbrt(5) * (zeta_3)^2) This isn't a very pretty way of representing the splitting field, but it is correct. Let F = Q(cbrt(5), cbrt(5) * zeta_3, cbrt(5) * (zeta_3)^2) It is in fact the case that F = Q(cbrt(5), zeta_3). And seeing this is really not hard at all. F is certainly contained in Q(cbrt(5), zeta_3) since fields are closed under multiplication. To see that Q(cbrt(5), zeta_3) is contained in F, we need to write cbrt(5) and zeta_3 as a combination of field operations involving only elements from F. This is quite easy: cbrt(5) = cbrt(5) and zeta_3 = cbrt(5) * zeta_3 / cbrt(5). This shows inclusion in both directions and establishes the fact that F = Q(cbrt(5), zeta_3). Thus the splitting field is Q(cbrt(5), zeta_3). I hope this helps. Please write back if you'd like to talk about this some more. - Doctor Paul, The Math Forum http://mathforum.org/dr.math/
Date: 06/13/2002 at 16:51:21 From: Lucia Subject: Thank you (Field theory-Abstract algebra) Dear Dr.Paul, Thank you for your complete and easy-to-understend answer. It was very helpful. Lucia
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