# Tagged Questions

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### Is $\mathbb{Z}[\sqrt{2} + \sqrt{3}]$ closed under multiplication?

Bonus question: if it's not, is it a subdomain of some ring of algebraic integers? This is just something I was thinking about a few weeks ago. I forgot about the concept of algebraic degrees, which ...
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### ideal calculations: $2\mathcal{O}_K=\mathfrak{B}^4$ in the ring of integers of $K=\mathbb{Q}(i,\sqrt{2m})$

Let $K=\mathbb{Q}(i,\sqrt{2m})$ where $m \in \mathbb{Z}$ is odd and squarefree. Let $\alpha = (1+i)\sqrt{2m}/2$. Then $\alpha^2=im$, such that $\alpha$ is part of the ring of integers $\mathcal{O}_K$. ...
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### Literature to the ring $\mathbb{Z}[\phi]$ where $\phi=\frac{1+\sqrt{5}}{2}$ is the golden ratio

I know few about algebraic number theory but recently I stumbled upon the ring $\mathbb{Z}[\phi]$ where $\phi = \frac{1+\sqrt{5}}{2}$ is the golden ratio. It seems to be a very interesting object to ...
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### Why study integrality?

Here are a few of the basic definitions related to integrality. (1) A polynomial in $R[x]$ is monic if its leading coefficient is $1$. (2) An element is integral over a ring $R$ if it ...
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### $(p)$ is a prime ideal in $\mathbb{Z}[\sqrt{n}]$ if and only if $n$ is not a square mod $p$

Let $n$ be a square-free integer such that $n\equiv 2,$ or $3\bmod 4$. If $p\in\mathbb{Z}$ be a prime such that $n$ is not a square modulo $p$, then $(p)$ is a prime ideal in ...
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### Factorize $(9+11\sqrt{-5})$ as a product of prime ideals in $\mathcal{O}_K$ where $K=\mathbb{Q}(\sqrt{-5})$

The ring of integers in this case, is $\mathbb{Z}[\sqrt{-5}]$. I have calculated that the norm of $(9+11\sqrt{-5})$ is $686=2\times 7^3$ and therefore its prime factorization must contain a prime ...
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### $\phi$ in $O_K$ but not in $\mathbb{Z}[t]$

I have this problem: Let $t$ be a root of the polynomial $f(x) = x³ + x² - 2x + 8$. Let $\phi = \displaystyle \frac{4}{t}$ and let $K = \mathbb{Q}(t)$. I was able to show that $f(x)$ is irreducible, ...
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### Computing the ring of integers of a number field

This question arose from the need to see the splitting behaviour of primes over an extension of number fields. One criterion is the Kummer's theorem, which gives the decomposition of the base prime in ...
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### Does any integral domain contain an irreducible element?

Let $R$ be an integral domain which is not a field. Does $R$ necessarily have an irreducible element? I suspect the answer is no, but I couldn't find an example showing that...
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### In general how to prove or disprove certain types of ideal?

i've come across a lot of questions recently that ask you whether or not there exist certain kinds of ideal, say; does there exist an ideal$J$of $\mathbb{Z}[i]$ for which $\mathbb{Z}[i] /J$ is a ...
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### Struggle proving maximal ideals principal in $\mathbb Z[\varphi]$

I was worried that my proof isn't right so I want to know if there are any mistakes in this and if this way can work? Thank you very much. We want to show every maximal ideal $\mathfrak m$ of ...
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### Infinitude of irreducibles in subring of an integer ring.

Let $\alpha \in \mathbb{C}$ be an algebraic integer of degree $n$, not a unit, and let $R = \mathbb{Z}[\alpha]$. Then every element $\beta \in R$ can be written uniquely in the form c_0+ c_1 ...
I want to complete the proof of the following theorem. Here is what I have got so far: Theorem Every non-euclidean valuation $v$ on a number field $K$ is equivalent to $v_{\mathfrak p}$ for some ...
What is the exact definition of a normalization of a Ring? I have to show this: normalization of multiplicative subset of domain And the answer already helped, but I don't know what $S^{-1}R'$ is ...