my question is , given the Group G of symmetries for the equation(adsbygoogle = window.adsbygoogle || []).push({});

[tex] x^{4} + a^{2}=0 [/tex]

for some 'a' Real valued i see this equation is invariant under the changes

[tex] x \rightarrow -x [/tex]

[tex] x \rightarrow ix [/tex]

[tex] x \rightarrow -ix [/tex]

[tex] x \rightarrow -x [/tex]

[tex] x \rightarrow i^{1/2}x [/tex]

[tex] x \rightarrow (-i)^{1/2}x [/tex]

under this symmetries we can see that we ONLY can have imaginary roots, since from the symmetries above any complex number solution to [tex] x^{4} + a^{2}=0 [/tex] should have an argument [tex] 4\phi = 2\pi [/tex] this is deduced from the base that [tex] x^{4} + a^{2}[/tex] is a real function for real 'x' , of course this example is TRIVIAL to prove to be true , but how about a more important case, could we deduce from my idea that ALL the roots of the function [tex] x^{-1}sinh(x)=0 [/tex] are ALL imaginary numbers ?

given any Polynomial K(x) with the following properties

* K(x) have ONLY pure imaginary roots (A)

* degre of K(x) is a multiple of '4' (B)

could we proof by any REDUCIBILITY theorem (over Real numbers) that the irreducible factors of K(x) over the field R are or will be of the form (the best possible chance) [tex] x^{4} + (a_i)^{2} [/tex] for some a_i ??

Another question is are conditions (A) and (B) equivalent ??

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# Symmetry group for roots

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