Re: quadratics with related roots

Your problem touches the essence of of Galois Theory.

Functions of the roots of a polynomial equation (p.e.) which are invariant under any permutation of the roots are known as symmetric functions.

Theorem: symmetric functions of the roots of a p.e. are expressible as rational functions of its coefficients.

In the case of y1 = x1/x2 and y2 = x2/x1, you have y1+y2 = (x1^2 + x2^2)/x1.x2 and y1.y2 = 1; y1+y2 is clearly symmetric in x1 and x2, so you are sure you can express y1+y2 rationally in the coefficients of the original quadratic equation.

A function of the roots of a p.e. which is invariant under even permutations of the roots and changes sign under odd permutations could be named "alternating function".

Theorem: alternating functions of the roots of a p.e. are expressible as rational functions of its coefficients and the square root of the discriminant.

This corresponds to the extension of the original coefficient field K to an encompassing field K(sqrt(D)), which is a 2D division algebra over K.

For a full treatment of Galois Theory see for instance Birkhoff and MacLane: A survey of modern algebra (MacMillan) - since its first edition in 1941 still on sale!

Happy studies: Johan E. Mebius

Adam Atkinson wrote:

I've been helping someone with maths, and some things have cropped
up which I remember seeing at school but have never used or seen since.

One particular item I'm curious about is questions of the form:

If the roots of the quadratic equation (insert quadratic here) are
alpha and beta, write down a quadratic whose roots are (two
formulas involving alpha and beta), without solving for alpha
and beta.

I've seen cases like asking for roots alpha^2 and beta^2
or alpha/beta and beta/alpha.

I did maths at university and don't recall ever having do to anything
like this. Is there some field of maths, science or engineering in
which people need to produce equations with related roots like this?



.

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