Let $p= x^2+ ax+b \in R[x] $ where $R= \mathbb{F}[a,b], a$ and $b$ are indeterminates over $F$.
It is mentioned that $p$ is a generic polynomial and it is irreducible, Since specialising the variables(a=0 and b=t) gives $\tilde p= x^2-t $ which is irreducible over $F[t][x]$.
My questions are
- What is the meaning of generic polynomial? In general, what is the meaning of the generic element?.
- how does the actual process of specialising variables work?
- why $\tilde p $ is irreducible?
Kindly help me with this. Thanks in advance
-I can't comment due to less points!
a generic polynomial refers usually to a polynomial whose coefficients are indeterminates.
You can use Gauss Lemma, Eisenstein's Criteria and Reduction Mod n method to find if, a given polynomial is irreducible or not.
we'll try with Reduction Mod n with example.. ${P= x^2-15x+7}$, we can take mod7 but it can't tell any thing about irreducibility over Z. so we we'll try mod 5.
P(x) mod5 = ${x^2+2}$ = ${x^2+b}$, now making a=0 it is a good choice to test irreducibility.
Now if it factor then we're left with (x+t1)(x+t2), so we need ${t1 \cdot t2 =2 =b}$, now,the perfect square in the ${Z_5} $ are {0,1,4}, but 2 doesn't appear.
Also [a,b] are field elements, hence it doesn't have zero devisors.
Hence if it is irrecucible over ${Z_n}$ then it is irreducible over ${Z}$
And Due to Gauss Lemma, if it is irreducible over Z, it is irreducible over Q.