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Please focus on the concept to solve this problem, because I can't handle to research on difficult terminology. Thanks in advance.

Find all real roots by Galois theory and find all other root to this equation: $x^8+x^6+x^4=340$

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closed as unclear what you're asking by Bookend, Solid Snake, USER91500, Strants, Aloizio Macedo Oct 4 '15 at 13:34

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question.If this question can be reworded to fit the rules in the help center, please edit the question.

@BrettFrankel - Problem is, i think the theory itself take Galois more than a year to create, This single problem may take 10 year for me to try and explore... – Victor Apr 24 '12 at 22:32
If you explain what you have tried so far, it will be easier for other users to find solutions at the appropriate level. – Brett Frankel Apr 24 '12 at 22:35
@BrettFrankel - i am trying to starting my reading to galois theory by asking this question, so to make sure i am on the right track. I don't understand what morphism and field/field extension, ring is, in addition i am not knowing what the irreducibilty really for... – Victor Apr 24 '12 at 22:40
@Victor: If you don't know what a ring is, you are not ready to tackle this question. You must start with the basics. – Zhen Lin Apr 24 '12 at 23:39
@Victor In order to understand Galois Theory you need to have a basic foundation of Abstract Algebra. Best way is to pick a book and read it thoroughly. If that doesn't work, try this . It is a really good lecture series by Benedict Gross, a Professor at Harvard. It is a pretty good place to start. If you prefer a book, I'd recommend Dummit and Foote. I am reading it right now and It is very good. Good Luck! – funktor Apr 25 '12 at 5:52
up vote 2 down vote accepted

If we can solve $$u^4+u^3+u^2=340$$ then by letting $u=x^2$ we can solve the original equation. But the displayed equation is of degree 4, and there is a formula for solving those. Just search for "quartic formula".

I'm sorry that this doesn't use Galois Theory, but I don't see what Galois Thoery can do for this problem.

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In fact, Galois "Theory" do not provides "method" to obtain roots of given equation although it is solvable, isn't it? – Groups Oct 4 '15 at 7:10
It is possible to use Galois Theory to solve a quartic, but unless you already know Galois Theory, it's the hard way. Some Galois Theory texts show you how to use it to solve cubics and quartics. – Gerry Myerson Oct 4 '15 at 12:22

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