I've been solving some problems preparing for my final exam of abstract algebra, and I want to know if my solution to this problem is correct, and ask some questions related to it too. The problem is:
Prove that the following polynomials are irreducible in $\mathbb{Q}[X]$:
(a) $\frac{2}{9}X^5 + \frac{5}{3}X^4+X^3+\frac{1}{3}$
(b) $2X^5-6X^3+9X^2-15$
(c) $X^3+6X^2+17+3$
(d) $X^4+4X^3+6X^2+2X+1$
This is the work I've done, tell me if there's anything wrong:
(a) For this one, I first multiply the polynomial by $9$, because $9\in\mathbb{Q}^\times$ and that means the result of the multiplication does not change the condition of being irreducible in $\mathbb{Q}[X]$. So the result of the multiplication is the polynomial $2X^5+15X^4+3X^3+3$, and I can use Eisenstein's criterion with $p=3$ to prove it's irreducible in $\mathbb{Q}[X]$.
(b) For this one, just Eisenstein's criterion with $p=3$ and it's concluded it's irreducible in $\mathbb{Q}[X]$.
(c) For this one, I started looking for possible roots in $\mathbb{Q}$. Since it's coefficients are all positive, the leading coefficient is $1$ and the independent coefficient is $3$, the only possible roots in $\mathbb{Q}[X]$ are $-1$ and $-3$. I evaluate the polynomial in both and see that it does not turn into $0$, so I conclude that this polynomial has no roots in $\mathbb{Q}[X]$, and then I conclude it's irreducible beacuse it's degree is $3$.
(d) For this last one, I use the fact that, given $p(X)$ irreducible, the map that turns $p(X)$ into $p(X+a)$ is homomorphism (hence it does not change the condition of being irreducible), so I calculate $p(X+1)$ and get as result $X^4+8X^3+24X^2+30X+14$, which I know is irreducible in $\mathbb{Q}[X]$ by Eisenstein's criterion with $p=2$.
So, given this work, the questions I have are:
Are my solutions correct?
In part (d), where I evaluated the polynomial in $(X+1)$, I did it randomly. I don't know how can I "decide the right $(X+a)$ to choose". Is there any tip or guide that may let me guess what's the right $(X+a)$ to then use Eisenstein's criterion?
I get confused with the "divisibility" concept in $\mathbb{Q}$. In integers its clear that, for example, $3$ divides $15$, but does $3$ divide $\frac{15}{2}$ in $\mathbb{Q}$? Or does $3$ divide $\frac{1}{2}$ in $\mathbb{Q}$? I'm not sure how this concept is defined in $\mathbb{Q}$.
Any help will be appreciated, thanks in advance.