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Questions tagged [integral-basis]

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

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Show that every $𝑥$ in $𝑂_𝐾$ has an even discriminant [duplicate]

Note: The discriminant $D$ of the cubic polynomial $x^3 + ax^2 + bx + c \in\mathbb Z[x]$ is $$D = a^2 b^2βˆ’4b^3βˆ’4a^3cβˆ’27c^2+18abc.$$ Let $K=\mathbb Q(\theta),$ where $πœƒ^3-πœƒ^2-2πœƒ-8=0.$ I can prove ...
Aseel .A's user avatar
5 votes
1 answer

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 ...
Username's user avatar
4 votes
1 answer

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 ...
Parcly Taxel's user avatar
2 votes
2 answers

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 \...
Infinity_hunter's user avatar
2 votes
2 answers

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 ...
mref's user avatar
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4 votes
2 answers

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 ...
Nikita Dezhic's user avatar
10 votes
1 answer

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 ...
Spook's user avatar
  • 4,948
1 vote
2 answers

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 ...
Conan Wong's user avatar
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6 votes
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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,...
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