# Don't understand Root objects as solutions of an equation

I am trying to solve an equation for a variable t under some assumptions.

However, the output produced is completely unclear to me. In particular, Mathematica's output says "Assuming a list of rules" and contains stuff like "#1^6 &" which I can't interpret (even though I understand the operators in general). Can anyone help? The expression I am trying to evaluate is as follows:

Assuming[
omega == 2 && sigma > 0 && sigma <= 1 && t > 0 && t <= 0.5 &&
sigma ∈ Reals && omega ∈ Reals && t ∈ Reals,
Solve[
(omega*sigma*(1/(omega*(t - 1)) + 1) - (sigma*t)/(t - 1)^2)/
(omega*sigma*t*(1/(omega*(t - 1)) + 1) + 1) - t/(t - 1)^2 -
1/(t - 1) - omega + log (omega*sigma*t*(1/(omega*(t - 1)) + 1) + 1)*
(omega*sigma*(1/(omega*(t - 1)) + 1) - (sigma*t)/(t - 1)^2) +
omega*(1/(omega*(t - 1)) + 1) -
2*omega^2*sigma*t*(1/(omega*(t - 1)) + 1)^2 +
(2*omega*sigma*t^2*(1/(omega*(t - 1)) + 1))/(t - 1)^2 == 0,
t]]


Any help would be very appreciated.

• Have you seen this? Let me also offer a tip: anything you see in your output that you don't understand, highlight it and press F1. – J. M.'s technical difficulties Sep 30 '17 at 14:40
• Your output involves Root objects. Think of them as a generalization of radicals, capable of representing any algebraic number. – John Doty Sep 30 '17 at 14:42
• Root[polynomial,n] is essentially the same kind of thing as Sqrt[x], but generalizes to cases where radicals can't go. – John Doty Sep 30 '17 at 15:08
• Also, did you mean to use log as a variable? The logarithm is Log[], using brackets, not parentheses. – John Doty Sep 30 '17 at 15:10
• Another tip for you. Since you're now (somewhat) familiar with Root[], look at the degree of the polynomial in its first argument. If it's 5 or greater, don't count on getting anything more explicit than that, per Abel and Galois. – J. M.'s technical difficulties Sep 30 '17 at 15:35

1. Mathematica has transformed your problem into finding the roots of a polynomial of degree 6 in t, which has 6 solutions. As always, it returns the solutions as a list of rules. In this case, each of the solutions is expressed as a Root object, because equations of degree 6 can usually not be solved in terms of radicals.

See Root in the documentation.

2. One of your terms is

log (omega*sigma*t*(1/(omega*(t - 1)) + 1) + 1)*
(omega*sigma*(1/(omega*(t - 1)) + 1) - (sigma*t)/(t - 1)^2)


The symbol log is being interpreted as a complex number. Is that intentional? Or were you trying to write an expression of form Log[...]? If the latter, then you problem is a simple syntax error.

• Thanks for your reply! No, it wasn't intentional. However, when I replace log by "Log[...]", the problem persists... – Alex Held Oct 1 '17 at 20:45