<s>[INST] Field extensions and irreducibility I'm having trouble trying to show that the function f=x^3 + x + 3 is irreducible in the rationals. I tried using Eisensteins criterion but it didn't work as it doesnt satisfy all conditions.
the second part of the question is to show that f has one real root which i showed using Rolle's Theorem from Analysis
the third part is showing that denoting the real root as x
Q(x)=Q(x^2) where Q(a) [represents the field extension over rationals]
My attempt at this question is to show that every member of Q(x) is contained in Q(x^2), by using closure under field operations, but I'm struggling to show that every member of Q(x^2) is inside Q(x).
Any help would be greatly appreciated.
 [/INST] Here is what I think for the first question:
Suppose $f(x) = x^{3} + x + 3$ is not irreducible over $\mathbb{Q}$.  Then we can write $f(x)$ as the product $(x + a)(x^{2} + bx + c)$ for some rational numbers $a, b, c$.
But then, multiplying this out gives the expression $x^{3} + bx^{2} + cx + ax^{2} + abx + ac = x^{3} + (b + a)x^{2} + (ab + c)x + ac$
And we know this must equal $x^{3} + x + 1$, so we know that:


*

*$b + a = 0$

*$ab + c = 1$

*$ac = 3$


But if a polynomial is irreducible over $\mathbb{Z}$, then it is irreducible over $\mathbb{Q}$, so we really only need to find a contradiction with the above equations if $a,b,c$ are integers.
Equation 3 means either $a = 1$, $c = 3$ or $a = 3$, $c = 1$, or $a = -1$, $c = -3$, or $a = -3, c = -1$.  But equation 2 implies $b = -a$, so:
If $a = 1$, $c = 3$, then $b = -1$, so from equation 2 we get $ab + c = 1(-1) + 3 = -1 + 3 = 2 \neq 1$. So equation 2 is not satisfied.
If $a = 3$, $c = 1$, then $b = -3$, and by equation 2 we get $3(-3) + 1 = -9 + 1 = -8 \neq 1$, so again equation 2 is not satisfied.
If $a = -1$, $c = -3$, then $b = 1$ and so $(-1)1 - 3 = -4 \neq 1$, which means equation 2 is yet again not satisfied.
Finally, if $a = -3$, $c = -1$, then $b = 3$, and so $(-3)3 - 1 = -10 \neq 1$, which means equation 2 is yet again not satisfied.
So in every case, we can't find a solution of integers that satisfies the three equations, and thus the polynomial is not reducible over $\mathbb{Z}$.  Consequently, it is not reducible over $\mathbb{Q}$.
</s>