On an old algebra prelim, there is a particular problem I would like some help on. It is a five-part question. Let $K$ be the splitting field over $\mathbb{Q}$ of the polynomial $$f(x)=x^5- x^4 + x^3 - x^2 +x -1 \text{.}$$ (a) What are the possible values for the minimum degree among the irreducible factors of a polynomial of degree $5$?
(b) Write $f$ as the product of factors irreducible over $\mathbb{R}$.
(c) Write $f$ as the product of factors irreducible over $\mathbb{Q}$.
(d) What is the degree of $K$ over $\mathbb{Q}$?
(e) What is the Galois group of $K$ over $\mathbb{Q}$?
Here's what I tried:
(a) Case 1: For a general quintic polynomial that is reducible (certainly $f$ factors, thanks to my TI-89), it may split into a product of a linear factor (degree $1$) and an irreducible quartic factor (degree $4$) $\textbf{or}$ a product of a linear factor and two irreducible quadratic factors (each of degree $2$) $\textbf{or}$ a product of two linear factors each of degree $1$ and an irreducible cubic factor (degree $3$).
Case 2: A quintic polynomial could also split into a product of three linear factors each of degree $1$ and an irreducible quadratic factor (degree $2$) $\textbf{or}$ a product of an irreducible cubic factor (degree $3$) and an irreducible quadratic factor (degree $2$).
Case 3: Finally, a quintic polynomial can also split into a product of five linear factors each of degree $1$ $\textbf{or}$ the quintic polynomial might already be irreducible of degree $5$.
I think a shorter answer would just be to consider the partitions of $5$, correct?
(b) I got the following for this: $$f(x)= (x-1)(x^2 -x +1)(x^2 +x +1) \text{.}$$
(c) Would this be the same as (b)?
(d) I'm not sure what it should be here, but I'm certain that $5$ is not the solution. Is the degree of the extension $10$?
(e) Having some difficulty with choosing the right automorphisms that permute the roots. At the moment, all I can say is that it will be a subgroup of $S_5$, is that right?
I would really appreciate all the help I can get.