I have no experience with this, so I dont know if this is too easy for MO.
Let $(a_n)$ be a strictly monotone sequence of natural numbers, then define the set of nice numbers of $(a_n)$ as $X(a_n):=\{ i |$there exists a choice of plus minus signs such that$ x^{a_i} \pm x^{a_{i-1}} \pm .... \pm x^{a_1} \pm 1$ is irredicible over $ \mathbb{Q} \}$.
Call a sequence pretty if $X(a_n)=\mathbb{N}$. For any pretty sequence one can define the set $Y(a_n)$ as the set of infinite 01-words such that the corresponding polynomials are irreducible (0 for choosing +1 and 1 for choosing -1).
For example, $a_n=n$ is pretty, see http://www.sciencedirect.com/science/article/pii/S0022404903002457 . What is $Y(a_n)$ in this example?
Question 0: Is there an example of a non-pretty sequence?
Question 1: Is the sequence $a_n$ pretty in one of the following cases and if yes what is $Y(a_n)$?:
-$a_n=F_n$ (Fibonacci-sequence starting with 1,2,3,5,...)
-$a_n=C_n$ (Catalan sequence starting with 1,2,5,...)
-$a_n$ the sequence of prime numbers.
Question 2: Does any subset of the set of infinite 01-words occur as a $Y(a_n)$?
Question 3: Are there examples where the sequence is pretty and the corresponding Galois groups are independent of the choice of $\pm$ ?
Question 4: Are there two pretty sequences with the same $Y(a_n)$ (and the same corresponding Galois groups)?
Question 5: Are the pretty sequences closed under some nice operations such as addition and multiplication? If yes, how do the $Y(a_n)$ behave under those operations.
Example to explain an element in $Y(a_n)$ at the Catalan numbers (assuming $X(a_n)= \mathbb{N}$, which I do not know):
x^2+1 is irreducible so the 01-sequence starts with 0, since we choose + first.
x^5+x^2+1 is irreducible so the sequence is now 00.
x^14-x^5+x^2+1 is irreducible so the sequence is now 001.
x^42-x^14-x^5+x^2+1 is irreducible so the sequence is now 0011.
x^132+x^42-x^14-x^5+x^2+1 is irreducible so the sequence is now 00110.
x^429-x^132+x^42-x^14-x^5+x^2+1 is irreducible so the sequence is now 001101.
