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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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    @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...2012-04-24
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    If you explain what you have tried so far, it will be easier for other users to find solutions at the appropriate level.2012-04-24
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    @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...2012-04-24
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    @BrettFrankel - May you help me, since without any example, starting reading a galois theory book would be difficult for me2012-04-24
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    @Victor: If you don't know what a ring is, you are not ready to tackle this question. You must start with the basics.2012-04-24
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    @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 http://www.extension.harvard.edu/open-learning-initiative/abstract-algebra . 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!2012-04-25

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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?2015-10-04
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    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.2015-10-04