Questions tagged [algebraic-numbers]
Use this tag for questions related to numbers that are roots of a non-zero polynomial in one variable with rational coefficients.
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On Roots of Non-Abelian Extentions of Q and the Abelian Closure of Q
This is somewhat nagging me. I just with to run it by some one cause it seems slightly unintuitive to me (in some regards).
Let us consider $r$ to be an algebraic number in $\mathbb{C}$ not in $\...
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Equivalent of complex analysis over the algebraic numbers
I was wondering what would happen if we wanted to do "complex-like" analysis but, instead, of $\mathbb{C}$, we would use the simplest (in terms of inclusion) characteristic $0$ algebraically ...
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Can three antiprisms (uniform polyhedra) fit exactly around an edge, leaving no gaps?
Let $\tau=2\pi$ $=360^\circ$.
An $n$-gon antiprism has dihedral angles
$$\theta_n = \arccos\left(-\frac1{\sqrt3}\tan\frac\tau{4n}\right)$$
(where an $n$-gon meets a triangle) and
$$\phi_n = 2\arccos\...
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Are there any generalizations of continued fractions to approximations of other polynomial equations? [closed]
One of the more interesting results about continued fractions, is that the continued fraction representation of a number repeats if and only if the number is a solution to a polynomial of degree 2 (or ...
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Criterion for subfield of $ \mathbb{C} $ to be dense
Question:
Is it true that a subfield $ K $ of $ \mathbb{C} $ is dense if and only if the roots of unity in $ K $ are dense in the unit circle?
Context:
I was thinking about the infinite degree ...
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Example of a non-reciprocal polynomial with nontrivial multiplicative relations between its roots
I'm trying to find an example of a polynomial with some special properties, or an idea of why there can't be one, if that's the case. Here's my setting:
Let $p(x)\in\mathbb{Z}[x]$ be an irreducible, ...
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Question about Rudin's PMA, Chapter 2 Exercise 2
I know that there are many questions related to this exercise that have been answered, and I have found a couple of different solutions to this exercise as well. But the solutions usually ignore the ...
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Rational dependences between algebraic numbers not closed under Galois conjugation [closed]
Let $p \in \mathbb{Z}[x]$ be an irreducible polynomial of degree $d > m > 1$ and $\alpha_1, \ldots, \alpha_m$ be a (strict) subset of all the distinct roots of $p$. Can the value of $c_1\alpha_1 ...
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Known "families" of algebraic numbers
Contrary to simple transcendental extensions of $\mathbb{Q}$, which are necessarily isomorphic to a field of rational functions over $\mathbb{Q}$, simple algebraic extensions are very varied - which ...
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Primes in quadratic rings that are not UFD's [duplicate]
In a quadratic ring $\mathbb{Z}[\sqrt{d}]$ that is not a UFD, is there a simple proof that if an element has a norm that is a prime integer, then that element of the ring is prime, not merely ...
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Primitive Elements and Bases of Fields [duplicate]
Good people!
I'm trying to prove a certain something, and I've reached a point where the whole thing will be complete if I can just prove the following lemma, which I'm actually not entirely certain ...
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is the degree of an algebraic number in a fixed algebraic number field bounded?
Let $K$ be an algebraic number field $K=\mathbb Q(\theta)$ with $\theta$ being a root of a minimal polynomial $f(x)$ of degree $n$ with rational coefficients. Let $\alpha$ be an arbitrary element of $...
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Is every complex root of an integer polynomial a product of an algebraic real number and a root of unity?
If $f \in\mathbb{Z}[x]$ and $u \in \mathbb{C}$ is a root of $f$, then do we always have that $u = a\xi$ where $a$ is some real algebraic number, and $\xi$ is some root of unity?
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Classification of linear algebraic subgroups of $GL(2,\overline{\mathbb{Q}})$
In the article "An algorithm determining the difference Galois group of second order linear difference equations" by Peter Hendriks from 1998, Lemma 4.1 is a consequence of, among other ...
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Guessing algebraic number by its binary digits
Positive integers $d$, $H$ are given. It's known that $r\in(0,1)$ is such that $p(r)=0$ for some nonzero $p\in\mathbb Z[x]$ with $\deg p\leq d$ and coefficients $c_i$, such that $|c_i|\le H$, $i=0,\...
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Is the statement of Baker's Theorem on wikipedia inaccurate?
The Wikipedia article on Baker's theorem states it as follows:
If $\lambda_1,\ldots,\lambda_n\in\mathbb{L}$ are linearly independent over the rational numbers, then for any algebraic numbers $\beta_0,...
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Which numbers can be a minimal value of a polynomial with integer coefficients?
Problem. Let $P$ be a polynomial with integer coefficients and consider $m(P) = \min_\limits{x\in\mathbb{R}} P(x)$. The problem is to describe all possible values of $m(P)$.
Necessary condition. If $m(...
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$p$-adic Algebraic Number Fields: Their Primes and Units
So I've been going through Kurt Hensel's old German classic Theorie der algebraischen Zahlen, and I've reached the point where he begins discussing $p$-adic algebraic number fields $\mathbb{Q}(p , \...
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Prove $\sqrt{2}+\sqrt{3}+\sqrt{5}+...+\sqrt{p_{n}}$ is irrational, where $p_{n}$ is the nth prime.
My motivation is making general proof , instead of trying to prove special cases.
To which branch of mathematics does my question belong?
I am highly interested in irrational numbers.
Is it good idea ,...
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Non-positively algebraic number
Let us say that an algebraic number $\alpha \in \mathbf{R}$ is non-positively algebraic if it is the root of a monic polynomial $p(x) = x^n + a_{n-1}x^{n-1} + a_{n-2}x^{n-2} + \dots + a_1 x + a_0$ ...
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Why does this entropy equation $H(p) = H(2p - p^2)$ have an algebraic solution?
The recent breakthrough paper of Justin Gilmer on the Union-Closed Set conjecture contains this mysterious coincidence:
Let
$$H(p) := - p \log p - (1 - p) \log (1- p)$$
be the entropy of a Bernoulli ...
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Why is the image of an element in $\mathbb Q(\zeta_m, \zeta_p)$ under an automorphism of $\mathbb Q(\zeta_m)$ defined?
This is a part of page 214 in Ireland and Rosen's "A Classical Introduction to Modern Number Theory". $p$ is a prime coprime to $m$, $\zeta_m = e^{\frac{2\pi i}m}$, $D_m$ is the ring of ...
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Why does $w^{p^n} - w \in P^2$ imply $w\in P^2$ for $P$ prime?
The corollary mentioned:
This is proposition 13.2.5 from Ireland and Rosen's "A Classical Introduction to Modern Number Theory". $D$ is the ring of algebraic integers of a cyclotomic field ...
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Computing the Minimal Polynomial of $\sqrt{3} + \sqrt[3]{5} $ over $\mathbb{Q} $
In trying to compute the minimal polynomial of $\sqrt{3} + \sqrt[3]{5} $ over $\mathbb{Q} $ I employed the usual approach of considering:
$x - (\sqrt{3} + \sqrt[3]{5} $) = 0
Then from there taking the ...
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Question about the proof that quotients of a ring of algebraic integers are finite
This is proposition 12.2.3 from Ireland and Rosen's "A Classical Introduction to Modern Number Theory". $D$ is the ring of algebraic integers of some algebraic number field, and $A$ is an ...
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Why does $P^e \subset (\alpha) + P^e$ imply that $(\alpha) + P^e$ is a power of $P$?
This is proposition 12.3.2 from Ireland and Rosen's "A Classical Introduction to Modern Number Theory". $D$ is the ring of algebraic integers of some algebraic number field, and $e$ is the ...
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Why is the order of an ideal with respect to a prime ideal well defined?
This is the definition of $\text{ord}_P A$ in Ireland and Rosen's "A Classical Introduction to Modern Number Theory". $D$ is the ring of algebraic integers of some algebraic number field and ...
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Why can we change the order of "exists" and "for all"?
This is lemma 5 in section 12.2 from Ireland and Rosen's "A Classical Introduction to Modern Number Theory". $D$ is the ring of algebraic integers of an algebraic number field $F$, and $N(x)$...
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Is my solution ok? The set of the algebraic numbers obtained as roots of polynomials with integer coefficients that have degree $n$ is countable.
I am reading "Understanding Analysis Second Edition" by Stephen Abbott.
The following exercise is Exercise 1.5.9(b):
Exercise 1.5.9(b) Fix $n\in\mathbb{N}$, and let $A_n$ be the algebraic ...
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What gives the golden ratio its unusual numerical properties?
Why does the golden ratio (and by extension the other metallic means) have such unusual numerical properties?
For those who don't know, the golden ratio ($\varphi$) is the positive root of the ...
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Understanding Liouville numbers and irrationality measure
Every number $x \in \mathbb{R}$ has an associated irrationality measure $\mu(x)$. Let $\mathbb{A}$ be the algebraic numbers and let $x_\mathbb{Q} \in \mathbb{Q}, x_{\mathbb{A}\setminus\mathbb{Q}} \in \...
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Problem in trigonometry solved with more advanced topics - $\displaystyle \cos(q\pi) \in \mathbb{Q} \to \cos(q\pi) \in \{0, \pm \frac{1}{2}, \pm 1\}$
Prove the following affirmation: If $q$ is a rational number and $\cos(q\pi)$ is also a rational number, prove that $\cos(q\pi)$ must be one of the elements of the set $\{0, \pm \frac{1}{2}, \pm 1\}$.
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Degree of $\sqrt[5]{2+\sqrt[3]{5+\sqrt{2}}} \cdot e^{2\pi i /3}$ as algebraic number
In this post, ‘degree’ means ‘degree as an algebraic integer’. Let $\alpha = \sqrt[5]{2+\sqrt[3]{5+\sqrt{2}}}$ I see that $f(\alpha)=0$ for the polynomial $f(x) = \big((x^{5}-2)^{3}-5)\big)^{2}-2$, ...
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Is every element of GF(p^k) whose degree is k a generator?
I'm reading notes on minimal polynomials and finite fields taken from Jim Belk's webpage
Proposition 7 goes:
It seems to me there exists a counterexample to this statement:
The number of elements of ...
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Algebraic independence of a family of numbers
I need to show that given $a_1,\cdots,a_n\in\mathbb{C}$ algebraic numbers linearly independent over $\mathbb{Q}$, then the numbers $e^{a_1},\cdots,e^{a_n}$ are algebraically independent over rationals....
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If $\alpha ,\beta$ algebraic over $F$ then there exist a isomorphism $\psi:F(\alpha) \to F(\beta) $iff $\alpha,\beta$ have same minimal polynomial
Let $K$ be a field extension of $F$ and $L$ be also a field extension
of F. If $\alpha \in K$ and $\beta \in L$ are both algebraic over $F$,
then show that there is an isomorphism of fields: $\mu : F(\...
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$F(\alpha)$ is isomorphic to the field $F(x)$ of rational functions over $F$ in the indeterminate $x$ where $\alpha$ is transcendental over $F$
Q.$(a)$ If $\alpha$ is transcendental over a field $F$, then show that
$(i)$ the map $\mu : F[x] → F[\alpha],f (x) \mapsto f (\alpha)$ is an
isomorphism; $(ii)$ $F(\alpha)$ is isomorphic to the field $...
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Let $K/F$ and $a,b \in K$ algebraic over $F$ and $[F(a):F]=m ,[F(b):F]=n$ then show that degree of $a + b, ab, a − b , ab^{−1}$ atmost $mn$ over $F$
Let $K$ be a field extension of the field $F$ and $a,b \in K$ be
algebraic over $F$.If a has degree $m$ over $F$ and $b \neq 0$ has
degree $n$ over $F$, then show that the elements $a + b, ab, a − b ,
...
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A question on the definition of algebraic [duplicate]
Let $K$ be a field extension of the field $F$. Then an element $\alpha$ of
$K$ is said to be
$(i)$ a root of a polynomial $f(x) = a_0 + a_1x +···+ a_nx^n$ in $F[x]$ iff $f (\alpha) = a_0 +a_1\alpha +··...
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Squares in Algebraic Extensions
Sorry if that's a hard question, but is there an "easy" way to determine when an element inside a (real) simple field extension of $\mathbb Q$, is a perfect square ? (The one in question is ...
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Is there a standard way of writing constructible numbers?
Every rational number can be represented by an irreducible fraction with a nonnegative denominator.
For expressions involving integers, the four basic operations of arithmetic and square roots, you ...
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Degree of elements in simple extension of $\mathbb{Q}$
Is the following statement true?
Let $x$ be algebraic over $\mathbb{Q}$ and let degree of $x$ over $\mathbb{Q}$ be n. Then, simple extension $\mathbb{Q}(x)$ will be equal to the set:
$$\{a_0 + a_1x + \...
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The discriminant and Stickelberger's Theorem
Let $\theta$ be a root of the polynomial $x^3-x+2$, which is irreducible. Consider the basis $\{1,\theta,\theta^2\}$, which is clearly a rational basis for the integer ring in $Q(\theta)$. Now, I'm ...
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positive(?) definition of a transcendental number - as opposed to negative def. not an algebraic number [closed]
Does this make sense? What's the formal definition of a transcendental number, but without saying it's not an algebraic number?
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Roots of polynomials with non-negative integer coefficients
Is there a way to describe/characterize all complex numbers $z$ with $|z| = 1$ such that $z$ is the root of some polynomial with non-negative integer coefficients?
For instance, I've found that such $...
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Any algebraic number with modulus $1$ is root of a polynomial with positive coefficients
Given a complex number on the unit circle $e^{i\theta}\neq 1$ that is the root of some polynomial in $\mathbb Z[x]$, can we always construct a polynomial $p(x)$ with positive integer coefficients such ...
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Which rational angles are constructable?
For which rational angles (I call an angle $\alpha$ rational if $\alpha/\pi \in \mathbb Q$) is $\sin(\alpha)$ algebraic with $[\mathbb Q(\alpha):\mathbb Q]$ being a power of 2? This condition is ...
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Find the integers of degree 4 extension [duplicate]
So, I have been trying to solve the following problem: find the ring of integers in $\mathbb{Q}(\sqrt{3}+\sqrt{7}).$
Of course, the answer for $\mathbb{Q}(\sqrt{3})$ is clear as is for $\mathbb{Q}(\...
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Generalizing algebraic numbers to multivariate polynomials
Consider the set of (complex) algebraic numbers $\mathbb{A}$, defined as $\mathbb{A} = \{x \in \mathbb{C} \mid p(x) = 0 \text{ for some $p \in \mathbb{Z}[x]$}\}.$ Suppose we now wanted to generalize ...
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Doubt regarding the two definitions of Norm of an ideal in Algebraic Number theory
Let $O_L$ and $O_K$ be Dedekind domains and Let $L$ and $K$ denote their corresponding field of fractions. Further, let $ L/K $ be a finite separable extension.
For a fractional ideal $J$ of $O_L$, we ...
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