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Abstract Algebra

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Irreducible Polynomial Over the Rationals (definition)

ABSALG-TGVUKE

Let $P=P(x)=a_nx^n+a_{n-1}x^{n-1}+\ldots+a_0\in{\mathbb Z}[x]$ be a polynomial of degree $n\ge 1$ with integer coefficients, that is $a_i\in{\mathbb Z}$ for $i=0,\ldots,n$ and $a_n\not=0$.

Which of the following is, by definition, equivalent to $P$ being irreducible over the field ${\mathbb Q}$ of rational numbers?​

A

There are no polynomials $Q$ and $R$ in ${\mathbb Q}[x]$ such that $P(x)=Q(x)R(x)$.

B

There are no non-zero polynomials $Q$ and $R$ in ${\mathbb Q}[x]$ such that $P(x)=Q(x)R(x)$.

C

If $P(x)=Q(x)R(x)$ for $Q$ and $R$ in ${\mathbb R}[x]$, then one of $Q$ or $R$ has degree 0.

D

If $P(x)=Q(x)R(x)$ for $Q$ and $R$ in ${\mathbb Q}[x]$, then one of $Q$ or $R$ has degree 0.

E

There is no $c\in{\mathbb Z}$ such that $P(x)=cQ(x)$ for a polynomial $Q$ in ${\mathbb Z}[x]$ whose coefficients have greatest common divisor $1$.