The discussion revolves around finding the splitting field of the polynomial x^4 - 7x in the complex numbers over the rational numbers. Participants are exploring methods to identify the roots of the polynomial and how to factor it effectively.
Some participants have provided insights into the polynomial's structure and suggested alternative methods for factoring, while others express confusion about the next steps in the process. There is an ongoing exploration of different interpretations and methods without a clear consensus on the best approach.
Participants are working under the constraints of the polynomial's degree and the requirement to find a splitting field, which may involve complex roots. There is also a mention of the challenge in identifying rational roots.
Is the best way to find all the roots in C to do what I've been doing? Assume an element is a root and then divide?mfb said:You can find all the roots (in C) and see how many of them are in Q.
PsychonautQQ said:Homework Statement
Hello PF. I need to find a splitting field of x^4-7x in C over Q
Homework Equations
The Attempt at a Solution
letting r be a root, I did the division and got x^4-7x = (x-r)(x^3+r*x^2+x*r^2+r^3). I'm a little confused on what to do now, do I just take another root and do the division again?
pasmith said:The right hand side is x^4 - r^4 which for fixed r is not identically equal to the left hand side for every x.
Starting with x^4 - 7x = (x - r)(x^3 + ax^2 + bx + c) and comparing coefficients of powers of x leads to <br /> a - r = 0, \\<br /> b - ar = 0, \\<br /> c - br = -7, \\<br /> cr = 0. This is a system in four unknowns a, b, c and r which has the solution a = r, b = r^2, c = r^3 - 7 and r(r^3 - 7) = 0. This of course gets you no closer to actually finding r.
Instead observe that x^4 - 7x = x(x^3 - 7) and then use the identity x^3 - r^3 \equiv (x - r)(x^2 + rx + r^2). That leaves you to factorize a quadratic.