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# 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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7 votes
2 answers
218 views

• 5,864
1 vote
0 answers
53 views

### 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 ...
0 votes
3 answers
65 views

### 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 ...
• 8,918
3 votes
0 answers
50 views

### 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, ...
• 51
4 votes
1 answer
210 views

### 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 ...
• 2,779
2 votes
1 answer
73 views

• 2,062
3 votes
0 answers
48 views

### 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 ...
• 101
2 votes
0 answers
70 views

• 434
5 votes
0 answers
82 views

• 2,052
1 vote
1 answer
95 views

### 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 ,...
• 1,554
1 vote
0 answers
96 views

### 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 $p(x) = x^n + a_{n-1}x^{n-1} + a_{n-2}x^{n-2} + \dots + a_1 x + a_0$ ...
• 203
0 votes
0 answers
73 views

### 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 ...
• 578
1 vote
0 answers
114 views

### 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 ...
• 331
0 votes
0 answers
35 views

### 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 ...
• 331
3 votes
2 answers
189 views

### 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 take the ...
1 vote
1 answer
49 views

### 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 ...
• 331
1 vote
1 answer
38 views

### 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 ...
• 331
1 vote
0 answers
129 views

### 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 ...
• 331
1 vote
0 answers
69 views

### 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 $N(x)$...
• 331
0 votes
0 answers
100 views

### 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 ...
• 8,790
1 vote
1 answer
179 views

### 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 ...
• 419
1 vote
0 answers
149 views

• 293
1 vote
2 answers
399 views

### $F(\alpha)$ 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 \$...
• 293