# How to solve this by Galois theory? [closed]

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 MacedoOct 4 '15 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 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! – funktor Apr 25 '12 at 5:52

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".