# Galois theory

31 views

### sobriquet

Nov 26, 2021, 10:19:04 PM11/26/21
to

Hey.

I've created this desmos demo to visualize some of the examples
given in the video.

https://www.desmos.com/calculator/bvzkej2b7w

Does anyone know of the method used to come up with these algebraic
equations associated with particular irrational numbers which
demonstrate how conjugate combinations match up?

Given a conjugate pair like {sqrt(2), -sqrt(2)}, how would you find
a suitable pair of expressions like {1/(3+2x), (1+x)^2} that are
invariant as you permute the conjugates?

### Brigdare Doss

Nov 27, 2021, 5:06:11 PM11/27/21
to
sobriquet wrote:

> Galois theory.

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Fall of the Kabal. Part 19
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### konyberg

Nov 27, 2021, 6:14:44 PM11/27/21
to
I might not understand what you want, but:
the equation that gives these two solutions are
(x - sqrt(2))(x - (-sqrt(2))) = (x - sqrt(2))(x + sqrt(2)) = x^2 - 2 = 0
KON

### sobriquet

Nov 27, 2021, 7:09:13 PM11/27/21
to
Did you watch the video?

x^2-2 equals 0, regardless whether you substitute sqrt(2) or -sqrt(2) for x.

But that is not what the video is about. The video is about algebraic expressions
that evaluate to the same result under the operation of swapping sqrt(2) and -sqrt(2)
as alternative substitutes for x.

So 1/(3+2x) is equal to (1+x)^2 if you substitute sqrt(2) for x in the first expression
and -sqrt(2) for x in the second expression and likewise if you substitute -sqrt(2)
for x in the first expression and sqrt(2) for x in the second expression.
But that doesn't mean that either of those expressions evaluates to the same
result regardless of whether you substitute sqrt(2) or -sqrt(2) for x.