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 [itex]x^4 - r^4[/itex] which for fixed [itex]r[/itex] is not identically equal to the left hand side for every [itex]x[/itex].
Starting with [itex]x^4 - 7x = (x - r)(x^3 + ax^2 + bx + c)[/itex] and comparing coefficients of powers of [itex]x[/itex] leads to [tex]
a - r = 0, \\
b - ar = 0, \\
c - br = -7, \\
cr = 0.[/tex] This is a system in four unknowns [itex]a[/itex], [itex]b[/itex], [itex]c[/itex] and [itex]r[/itex] which has the solution [itex]a = r[/itex], [itex]b = r^2[/itex], [itex]c = r^3 - 7[/itex] and [itex]r(r^3 - 7) = 0[/itex]. This of course gets you no closer to actually finding [itex]r[/itex].
Instead observe that [itex]x^4 - 7x = x(x^3 - 7)[/itex] and then use the identity [itex]x^3 - r^3 \equiv (x - r)(x^2 + rx + r^2)[/itex]. That leaves you to factorize a quadratic.
A splitting field is a field extension that contains all the roots of a given polynomial. In other words, it is the smallest field in which a polynomial can be completely factored into linear factors.
To find the splitting field of a polynomial, you need to factor the polynomial into irreducible factors and then find the smallest field extension that contains all the roots of those factors. In the case of x^4-7x, the irreducible factors are x and x-7, so the splitting field would be the smallest field extension of Q that contains the roots of x and x-7.
Finding the splitting field is important because it allows us to fully understand the behavior of a polynomial. It also helps us solve equations involving the polynomial and determine its Galois group, which provides valuable information about the solvability of the polynomial.
The splitting field of x^4-7x in C over Q is the field extension of Q that contains all the roots of x^4-7x. In this case, the splitting field would be the field extension of Q that contains the roots of x and x-7, which is C.
Yes, the splitting field of x^4-7x is unique. This is because the splitting field is the smallest field extension that contains all the roots of the polynomial, and the roots of x^4-7x are x and x-7. Since the field extension of Q that contains these roots is unique (C in this case), the splitting field is also unique.