Edit:There were several major mistakes by my side this post, most of which have been accounted for.Now, after editing these out, the post seems to have no purpose at all.Nevertheless, it feels wrong to delete it, so I am going to leave it as it is.
Consider the splitting field of $$x^3-2$$ which is $$K=Q(\alpha,\omega),$$ where alpha is the real cube root of 2 and omega is a primitive third root of unity.One can check that in fact $$K=Q(\alpha+\omega).$$ The galois group of K over Q is isomorphic to $$Z_2\times Z_3$$ and is abelian, so , by the Kronecker-Weber theorem, lies in a cyclotomic field, the smallest of which is dependent on the conductor of K.EDITThis in fact is wrong, the galois group of this extension is nonabelian, so K doesn't lie in a cyclotomic field.
I was trying to solve $$a^3+2b^3+4c^3=1$$ in integers, which is the norm of an element of the form $$a+b\alpha+c\alpha^2.$$EDIT The norm is actually $$a^3+2b^3+4c^3-6abc$$Let $$O_K$$ be the ring of integers, so $$Z(\alpha,\omega)$$ is contained in it.By Dirichlet's unit theorem, the ring of integers is generated by 2 elements.EDIT the fundamental units can be found here:http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/unittheorem.pdf
The diophantine equation seems to have a lot of solutions:(1,0,0),(5,-4,1),(-1,1,0) etc.So to solve this, we have to see when an element of the previous form is a product of powers of the two units.But the fundamental units look terrifying, so maybe this won't be a very fruitful process.
Also, if someone can tell me what the actual ring of integers is, that would be really helpful.