Determine if sqrt(-3) is an element in a splitting field

Thread starter gop
Start date May 15, 2010
Tags

Element Field Splitting

AI Thread Summary

The discussion revolves around determining whether sqrt(-3) is an element of the splitting field L generated by the polynomial x^2+x+1 over the rational numbers, as well as whether it belongs to the field Q(a), where a is a complex root of x^3+x+1. Participants suggest starting the solution by finding the roots of the polynomials involved. The complexity of the problem arises from the nature of the roots and their relationship to the elements in the fields. Ultimately, the discussion emphasizes the importance of understanding polynomial roots in relation to field extensions.

May 15, 2010

gop

Homework Statement

L is the splitting field generated by x^2+x+1 (over \mathbb{Q})
a) Is \sqrt{-3} an element of L?
b) Is sqrt(-3) an element of \mathbb{Q}(a), where a is a complex root of x^3+x+1?

Homework Equations

The Attempt at a Solution

Really no idea.

Physics news on Phys.org

May 15, 2010

VeeEight

You should start by finding the roots of the given polynomials.

Thread 'Finding the number of ways to arrange identical balls in a circle (3 different colors)'

I tried to combine those 2 formulas but it didn't work. I tried using another case where there are 2 red balls and 2 blue balls only so when combining the formula I got ##\frac{(4-1)!}{2!2!}=\frac{3}{2}## which does not make sense. Is there any formula to calculate cyclic permutation of identical objects or I have to do it by listing all the possibilities? Thanks

View full post »

Thread 'Greatest possible value of a constant in polynomial'

Since ##px^9+q## is the factor, then ##x^9=\frac{-q}{p}## will be one of the roots. Let ##f(x)=27x^{18}+bx^9+70##, then: $$27\left(\frac{-q}{p}\right)^2+b\left(\frac{-q}{p}\right)+70=0$$ $$b=27 \frac{q}{p}+70 \frac{p}{q}$$ $$b=\frac{27q^2+70p^2}{pq}$$ From this expression, it looks like there is no greatest value of ##b## because increasing the value of ##p## and ##q## will also increase the value of ##b##. How to find the greatest value of ##b##? Thanks

View full post »