All Questions
3
questions
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Factorizaton in an Euclidean ring
I have a doubt concerning Lemma 3.7.4 from Topics in Algebra by I. N. Herstein.
The statement of the Lemma is:
Let $R$ be a Euclidean ring. Then every element in $R$ is either a unit in $R$ or can be ...
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3
votes
1
answer
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Problem in understanding the unique factorization theorem for Euclidean Rings.
Unique Factorisation Theorem: Let $R$ be a Euclidean ring and $a\neq 0$ non-unit in $R.$ Suppose that $a =\pi_1\pi_2\cdots\pi_n=\pi_1'\pi_2'\cdots\pi_m'.$ where the $\pi_i$ and $\pi_j'$ are prime ...
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Prime elements in the ring of integers adjoin a complex number.
Let $R=\mathbb Z[e^{i\pi/3}]=\{a+be^{i\pi/3}\mid a,b\in \mathbb Z\}\subseteq \mathbb C$
(a) Show that $R$ is a Euclidean domain using the Euclidean norm $N(u)=|u|^2$.
(b) Show that if $p$ is a prime ...
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