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Abstract Algebra

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Moderate

Polynomial Rings:Properties $F[x,y]$ shares with $F[x]$,$F$ field

ABSALG-PWFFCD

Let $F$ be a field, let $F[x]$ be the ring of polynomials in one variable $x$ with coefficients in $F$, and let $F[x,y]$ be the ring of polynomials in two variables $x$, $y$ with coefficients in $F$.

Below, we list some true properties of the ring $F[x]$. Select all the complete sentences that describe properties of $F[x]$ that are not shared by $F[x,y]$.

Highlight Answer(s) Below

$F[x]$ is an Integral Domain. $F[x]$ is a Unique Factorization Domain. $F[x]$ is a Principal Ideal Domain. $F[x]$ is a Euclidean Domain. The units of $F[x]$ are the nonzero constant polynomials.