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Questions tagged [integral-basis]

For questions about integral basis, a concept in algebraic number theory

Let $K$ be a number field of degree $n$, $O_K$ be the ring of integers of $K$ and $I$ be an ideal of $O_K$. A basis for ideal $I$ is the set of elements $\{a_1, a_2 , \ldots, a_n \}$ such that every element of $I$ can be written uniquely as $$\sum_{j=1}^{n}x_ja_j$$ where $x_j\in \mathbb{Z}$.

An integral basis is a basis for $O_K$.

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Find an integral basis of $\mathbb{Q}(\alpha)$ where $\alpha^3-\alpha-4=0$

Let $K=\mathbb{Q}(\alpha)$ where $\alpha$ has minimal polynomial $X^3-X-4$. Find an integral basis for $K$. I have calculated the discriminant of the minimal polynomial is $-2^2 \times 107$, so the ring of algebraic integers is contained in…
Spook
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Alternative integral basis for $\Bbb{Z}[w]$

Write $w= e^{2\pi i/m}$ for $m \geq 3$. Consider the number field $K = \Bbb{Q}(\omega)$ and the ring of integers $\mathcal{O}_K = \Bbb{Z}[w]$ that has the usual integral basis $$B = \{1,w,w^2,\ldots,w^{\varphi(m) - 1}\}$$ where $\varphi$ is the…
user38268
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Integral Basis of $O_k$

Let $K=Q(\sqrt 6,\sqrt{11})$. Write $α ∈ O_K$ and its conjugates in terms of a $Q$-basis. And show that an integral basis of $O_K$ is given by ${1,\sqrt 6,\sqrt {11},\frac{\sqrt 6+\sqrt{66}}2 }$, from first principles. I'm really not sure how to do…
Username
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Guaranteed Integral Basis for Pure (Cubic) Fields

For quadratic fields $\mathbb{Q}(\sqrt{d})$, with $d$ squarefree, we know that an integral basis is \begin{align*} \begin{cases} 1, \frac{1 + \sqrt{d}}{2} & d = 1 \mod 4, \\ 1, \sqrt{d} & d = 2, 3 \mod 4, \end{cases} \end{align*} while for pure…
Colin Jackson
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On the representation of $\sqrt{\pm p}$ in the integral basis of $\mathbb Q(\zeta_p)$

I took another look at my previous question on proving a certain trigonometric identity related to the braced heptagon: $$\sin\frac\pi7-\sin\frac{2\pi}7-\sin\frac{4\pi}7=-\frac{\sqrt7}2$$ Around the time I posted the first question I was trying to…
Parcly Taxel
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Is field-basis of integral elements an integral basis?

Suppose we have a finite field extension $K = \mathbb{Q(\alpha)}$ with basis $1,\alpha,\dots,\alpha^{n-1}$ where all $\alpha^i$ are integral elements. Do they form an integral basis of the ring of integers $\mathcal{O}_K$ of $K$?
Nikita Dezhic
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Integral basis for $\mathbb{Q}(\theta)$ where $\theta^3 - 4 \theta + 2 =0$

I am trying to find an integral basis for the field $K =\mathbb{Q}(\theta)$ where $\theta^3 - 4 \theta + 2 =0$. I suspect that $\{1, \theta, \theta^2 \}$ is a potential candidate. For $a,b,c \in \mathbb{Z}$ it is easy to see that $a + b\theta +…
Infinity_hunter
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Integral Basis of $\mathbb{Q}(\sqrt{i})$

I understand the Integral basis of $\mathbb{Q}(\sqrt{i})$ is: $\{1,i,\sqrt{i},i\sqrt{i}\}$ . To find out this basis, have i each time to compute the discriminant? Any reference book or notes where explains it in a simple way? Thanks
mref
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Representation of conjugates of primitive algebraic integer in cyclic Galois field extension

Let $K/\mathbb{Q}$ be a cyclic Galois extension of degree $n$ where $K=\mathbb{Q}(\theta)$ and $\theta$ is an algebraic integer. How can I describe the conjugates of $\theta$ in terms of the $\mathbb{Q}$-basis…
Ama
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Algebraic Number Theory - Integral Basis

Let $K$ be a number field with $[K:Q] =n$. Let $O_k$ be its ring of algebraic integers. I understand how there is an integral basis for $Q$, i.e. $\exists$ a $Q$-basis of $K$ consisting of elements of $O_k$. Let this integral basis be denoted by…
Conan Wong
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Integral basis for multiquadratic number field

Let $K=\mathbb{Q}(\sqrt{a_1},\ldots,\sqrt{a_r})$ for square-free integers $a_1,\ldots,a_r$ --a multiquadratic number field. Suppose the generating elements $a_1,\ldots,a_r$ is normalized. For each integer $j\in\{1,\ldots,2^r\}$, there is a unique…
HRDelaCruz
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