Yes that makes sense that its a Galois extension, because we're told that each of the roots are distinct (and there are n of them) so it is a splitting field and has characteristic 0, thus it's Galois. Although your argument makes perfect sense, saying that |GalQQ(z)|=[Q(z):Q]=1 if the group is...
I am trying to show that if z, z2, z3, ..., zn=1 are n distinct roots of xn-1 in some extension field of Q (the rationals), then GalQQ(z) (the galois group of Q(z) over Q) is abelian. Would I be wrong to say that since the galois group we're talking about here only involves an extension field...
I am trying to prove that if c is a root of f(x) in Z_p then c^p is also a root. It seems very simple but I can't think how to approach it. Any insight on this would be greatly appreciated, and sorry for not using the latex but it seems to be acting up.
I'm currently trying to prove that (for a field extension K of the field F) if u\in K and u^2 is algebraic over F then u is algebraic over K.
I thought of trying to prove it as contrapositive but that got me nowhere--it seems so simple but I don't know what to use for this. Any help with this...
I'm trying to figure out how to prove that every polynomial in \mathbb{Z}_9 can be written as the product of two polynomials of positive degree (except for the constant polynomials [3] and [6]). This basically is just showing that the only possible irreducible polynomials in \mathbb{Z}_9 are the...
I am trying to prove that the additive groups \mathbb{Z} and \mathbb{Q} are not isomorphic. I know it is not enough to show that there are maps such as, [tex]f:\mathbb{Q}\rightarrow \mathbb{Z}[/itex] where the input of the function, some f(x=\frac{a}{b}), will not be in the group of integers...
I'm trying to figure out how to prove the following...
If a, b \in G where G is a group, then the order of bab^{-1} equals the order of a.
I'm rather stumped because the group is not necessarily abelian and it seems like it would have to be in order to directly show that you can rearrange...
Hello, I'm working out of Hungerford's Abstract Algebra text and this proof has been bothering me because I think I know why it works and it's so simple but I can't figure out how you would show a rigorous proof of it...
If a=p_1^{r_1}p_2^{r_2}p_3^{r_3} \cdots p_k^{r_k} and...