| | $\mathbb{Z}$ ([[Integers]]) | $F[x]$ ([[Ring of Polynomial Forms]] over a [[Field (Algebra)\|field]] $F$) | | ----------------------- | ------------------------------------------------------------------------------------------------- | --------------------------------------------------------------------------------------------------------------------------------- | | Form of Elements: | $a_{n}10^{n}+a_{n-1}10^{n-1}+\dots +a_{0}$ | $a_{n}x^{n}+a_{n-1}x^{n-1}+\dots a_{1}x+a_{0}$ | | Euclidean Domain: | $v(a)=\|a\|$ | $v(f)=\deg(f)$ | | Units: | $a$ is a unit iff $\|a\|=1$ | $f$ is a [[Units of Ring of Polynomial Forms over Integral Domain\|unit]] iff $\deg(f)=0$ | | Division with remainder | [[Division with remainder for integers]] | [[Division with Remainder Theorem for Ring of Polynomial Forms over Fields]] | | PID: | Every nonzero [[Ideals of Integers\|ideal]] $I=n\mathbb{Z}$ where $n\neq 0$ and $\|n\|$ is minmum | Every nonzero [[Ring of Polynomial Forms over Field is a Principal Ideal Domain\|ideal]] $I=f(x)F[x],$ where $\deg(f)$ is minimum | | Irreducible Elements: | [[Prime numbers]] only factors are $1$ and itself | [[Irreducible Polynomial]] only factors are units and associates | | Bézout's lemma | [[Bézout's lemma]] | [[Bézout's Identity for Ring of Polynomial Forms Over Field]] | | Euclid's lemma | [[Euclid's lemma]] | [[Euclid's Lemma for Irreducible Polynomial Forms over Field]] | | UFD: | [[Fundamental theorem of arithmetic]] | [[Ring of Polynomial Forms over Field is Unique Factorisation Domain]] |