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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 Normal Human, Solid Snake, Deutsch Mathematiker, Strants, Aloizio Macedo Oct 4 at 13:34

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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... – 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! – dashdart Apr 25 '12 at 5:52

1 Answer 1

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 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 at 12:22

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