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

Thread starter gop
Start date May 15, 2010
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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.

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May 15, 2010

VeeEight

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

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