Galois group of x^3-3x+3

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What is the Galois group of x^3-3x+3? EDIT: The method used in this video does not show that the polynomial has only a single real root since we didn't test for where the polynomial is increasing/decreasing. To see why it only has one root in R we can either observe that at both critical points the function is positive (so it can't have a root between these points, since then the derivative would be 0 at some point between -1 and 1), or we could use the first derivative to see where the function is increasing/decreasing.

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30 сен 2024

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Комментарии : 13

@cmilkau 5 месяцев назад

you need to calculate f *AT* the critical points (and ±∞), not around them, in order to determine the graph shape of the polynomial ...

@coconutmath4928 5 месяцев назад

Yes, there are some details missing from that part of the video... hopefully it is still helpful :O

@jyotiprakashchowdhury7252 8 месяцев назад

A short introduction to Galois Group will be highly appreciated

@shortstoriesglenrose4382 6 месяцев назад

For such a small example it doesn't matter, but generally speaking transitivity is a weaker property for a group action than the existence of an n-cycle. A simple example is A_4, which certainly crops up as the Galois group for various quartics.

@coconutmath4928 5 месяцев назад

Yes that's true... transitivity only guarantees a p-cycle when p is prime.

@vasil_astro_go 3 месяца назад

Thank you!

@juliefinkjulesheartmagic1111 Год назад

Thanks for another great explanation!

@meiliyinhua7486 Год назад

When testing for where it crosses the x axis, why not pick the critical points to test? Since we know the left and right infinite limits based on the leading term, all we need to do to test for crossings is find the location of maxima and minima where f(1) = 1 and f(-1) = 5 Meanwhile the testing points chosen do not rule out the possibility of a negative value between 0 and 2 until you combine it with the value of f(1)

@coconutmath4928 Год назад

That is a good point, I forgot to do my first derivative test properly haha. I edited the description with the method you were saying.