The Quadratic Integer Ring $\Z[\sqrt{5}]$ is not a Unique Factorization Domain (UFD)

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• The Quadratic Integer Ring $\Z[\sqrt{-5}]$ is not a Unique Factorization Domain (UFD) Prove that the quadratic integer ring $\Z[\sqrt{-5}]$ is not a Unique Factorization Domain (UFD). Proof. Any element of the ring $\Z[\sqrt{-5}]$ is of the form $a+b\sqrt{-5}$ for some integers $a, b$. The associated (field) norm $N$ is given […]
• The Ideal Generated by a Non-Unit Irreducible Element in a PID is Maximal Let $R$ be a principal ideal domain (PID). Let $a\in R$ be a non-unit irreducible element. Then show that the ideal $(a)$ generated by the element $a$ is a maximal ideal.   Proof. Suppose that we have an ideal $I$ of $R$ such that \[(a) \subset I \subset […]
• Ring of Gaussian Integers and Determine its Unit Elements Denote by $i$ the square root of $-1$. Let \[R=\Z[i]=\{a+ib \mid a, b \in \Z \}\] be the ring of Gaussian integers. We define the norm $N:\Z[i] \to \Z$ by sending $\alpha=a+ib$ to \[N(\alpha)=\alpha \bar{\alpha}=a^2+b^2.\] Here $\bar{\alpha}$ is the complex conjugate of […]
• A ring is Local if and only if the set of Non-Units is an Ideal A ring is called local if it has a unique maximal ideal. (a) Prove that a ring $R$ with $1$ is local if and only if the set of non-unit elements of $R$ is an ideal of $R$. (b) Let $R$ be a ring with $1$ and suppose that $M$ is a maximal ideal of $R$. Prove that if every […]
• 5 is Prime But 7 is Not Prime in the Ring $\Z[\sqrt{2}]$ In the ring \[\Z[\sqrt{2}]=\{a+\sqrt{2}b \mid a, b \in \Z\},\] show that $5$ is a prime element but $7$ is not a prime element.   Hint. An element $p$ in a ring $R$ is prime if $p$ is non zero, non unit element and whenever $p$ divide $ab$ for $a, b \in R$, then $p$ […]
• The Ring $\Z[\sqrt{2}]$ is a Euclidean Domain Prove that the ring of integers \[\Z[\sqrt{2}]=\{a+b\sqrt{2} \mid a, b \in \Z\}\] of the field $\Q(\sqrt{2})$ is a Euclidean Domain.   Proof. First of all, it is clear that $\Z[\sqrt{2}]$ is an integral domain since it is contained in $\R$. We use the […]
• The Quotient Ring $\Z[i]/I$ is Finite for a Nonzero Ideal of the Ring of Gaussian Integers Let $I$ be a nonzero ideal of the ring of Gaussian integers $\Z[i]$. Prove that the quotient ring $\Z[i]/I$ is finite. Proof. Recall that the ring of Gaussian integers is a Euclidean Domain with respect to the norm \[N(a+bi)=a^2+b^2\] for $a+bi\in \Z[i]$. In particular, […]
• Three Equivalent Conditions for an Ideal is Prime in a PID Let $R$ be a principal ideal domain. Let $a\in R$ be a nonzero, non-unit element. Show that the following are equivalent. (1) The ideal $(a)$ generated by $a$ is maximal. (2) The ideal $(a)$ is prime. (3) The element $a$ is irreducible. Proof. (1) $\implies$ […]