# Tagged Questions

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### Ring of integers for $\mathbb{Q}(\sqrt{23},\sqrt{3})$

What is the ring of integers for $\mathbb{Q}(\sqrt{23},\sqrt{3})$? So, these are numbers of the form $a+b\sqrt{3}+c\sqrt{23}+d\sqrt{69}$ where $a,b,c,d\in\mathbb{Q}$, and we want to find ones whose ...
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### Roots of $x^n - 1$ in an algebraically closed field of prime characteristic

Let $F$ be an algebraically closed field of characteristic $p$ , and let $n$ be a positive integer. Consider $g := x^n - 1 \in F[x]$ Is it true that $g$ has distinct roots in $F$ if and only if ...
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### Ring Isomorphism Proof

Let $p$ be a prime with $p \equiv 1 (\mod 4 )$. I am trying to show that $\mathbb{Z}[X]/(X^2 + 1, p) \cong \mathbb{Z}_p \times \mathbb{Z}_p$ is a ring isomorphism. I am not really sure how to ...
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### Show that if $c_1 + c_2\sqrt{5}$ divides $n$ in ${\bf{O}}[\sqrt{5}]$, then so does $c_1 - c_2\sqrt{5}$

I have a ring: $${\bf{O}}[\sqrt{5}] = \{c_1 + c_2\sqrt{5}: (c_1 \in \mathbb{Z} \wedge c_2 \in \mathbb{Z}) \lor (c_1 + \frac{1}{2} \in \mathbb{Z} \wedge c_2 + \frac{1}{2} \in \mathbb{Z}) \}.$$ I ...
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### Whether a domain is Dedekind or not

We know from algebraic number theory that if $d$ is a square-free integer, $d\neq 0,1$ and $d$ is congruent to $2,3$ modulo $4$ then $\mathbb{Z}[\sqrt{d}]$ is the ring of integers of the quadratic ...
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### Probability that two Gaussian integers are divisible

Let $z = x+ iy \in \mathbb{Z}[i]$ and let $a+ib \in \mathbb{Z}[i]$ with $a^2 + b^2 \equiv 1 \mod{4}$. What is the probability that $a+ib$ divides $x + iy$ in $\mathbb{Z}[i]$? This question would ...
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### Probability that $x \equiv 3 \pmod{4}$

I am working on a number theory project and I am interested in the following statement: What is the probability that an integer $x$ has the property that $|x| \equiv 3 \mod{4}$? This seems ...
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### Non-commutative or commutative ring or subring with $x^2 = 0$

Does there exist a non-commutative or commutative ring or subring $R$ with $x \cdot x = 0$ where $0$ is the zero element of $R$, $\cdot$ is multiplication secondary binary operation, and $x$ is not ...
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### Is there an explicit embedding from the various fields of p-adic numbers $\mathbb{Q}_p$ into $\mathbb{C}$?

For any field of p-adic numbers $\mathbb{Q}_p$, one can construct the field $\mathbb{C}_p$, the metric completion of one of its algebraic completions. By the axiom of choice, we can prove this to be ...
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### An ideal, $I$, is maximal iff $R/I$ is a field

The first line of the proof given in my book says that the ideals of $R/I$ are in bijective correspondence with the ideals of $R$ lying between $I$ and $R$. What is the bijection?
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### Integral extensions

Let $p\neq1$ be an integer and let $\beta$ be a root of $x^6-p$. What is the difference, in terms of $\mathbb{Z}$-modules, between $\mathbb{Z}[\beta]$ and $\mathbb{Z}[\beta^2,\beta^3]$? I can ...
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