Algebra 1 - Tutorium (25/26)

Decide which of the following polynomials are irreducible:

1. \(X^3 + 3X + 2 \in \mathbb{F}_5[X]\)
   
2. \(X^4 - 2X^2 - 8 \in \mathbb{Q}[X]\)
   
3. \(-20 X^7 + 21X^3 - 7 X^2 + 28 \in \mathbb{Q}[X]\)
   
4. (*) \(2X^4 - 8X^2 + 1 \in \mathbb{Q}[X]\)
   
5. (*) \(X^2 + 6X + 10 \in \mathbb{Z}[X]\)
   

prove the criterion about inverting coefficients:
Let $f = α_n Xⁿ + … + \(\alpha_0\).

TFAE:

1. \(f\) is irreducible
   
2. \(\sum_{i=0}^n \alpha_{n-i} X^i\) is irreducible
   

Hint:
consider the map

\begin{align*} \mathrm{inv}: R[X] \rightarrow& R[X] \\ \sum_{i=0}^n \alpha_i X^i \mapsto& \sum_{i=0}^n \alpha_{n-i} X^i \end{align*}

and show that it is a monoid isomorphism of \((R[X],\cdot)\).

prove the rational root theorem:

Let \(R\) be an UFD, \(f = \sum_{i=0}^{n} \alpha_i X^i \in R[X]\) a polynomial.
Suppose \(p,q\) are coprime elements, such that in the quotient ring

\begin{align*} f(\frac{p}{q}) = 0 \end{align*}

Then it follows, that

\begin{align*} p \mid& \alpha_0 \\ q \mid& \alpha_n \end{align*}

find all irreducible polynomials in \(\mathbb{F}_2[X]\) with \(\mathrm{deg}(f) \leq 3\) (or \(\mathrm{deg}(f) \leq 4\))