All Questions
Tagged with real-numbers abstract-algebra
21
questions
76
votes
7
answers
33k
views
Is an automorphism of the field of real numbers the identity map?
Is an automorphism of the field of real numbers $\mathbb{R}$ the identity map?
If yes, how can we prove it?
Remark An automorphism of $\mathbb{R}$ may not be continuous.
- 42.9k
9
votes
4
answers
2k
views
Using Zorn's lemma show that $\mathbb R^+$ is the disjoint union of two sets closed under addition.
Let $\Bbb R^+$ be the set of positive real numbers. Use Zorn's Lemma to show that $\Bbb R^+$ is the union of two disjoint, non-empty subsets, each closed under addition.
- 517
13
votes
2
answers
532
views
Is there an "algebraic" way to construct the reals?
It's possible to construct $\mathbb{Q}$ from $\mathbb{Z}$ by constructing $\mathbb{Z}$'s field of fractions, and it's possible to construct $\mathbb{C}$ from $\mathbb{R}$ by adjoining $\sqrt{-1}$ to $\...
- 694
14
votes
2
answers
2k
views
Finite dimensional division algebras over the reals other than $\mathbb{R},\mathbb{C},\mathbb{H},$ or $\mathbb{O}$
Have all the finite-dimensional division algebras over the reals been discovered/classified?
The are many layman accessible sources on the web describing different properties of such algebras, but ...
- 940
9
votes
2
answers
1k
views
Proving (without using complex numbers) that a real polynomial has a quadratic factor
The Fundamental Theorem of Algebra tells us that any polynomial with real coefficients can be written as a product of linear factors over $\mathbb{C}$. If we don't want to use $\mathbb{C}$, the best ...
- 23.7k
10
votes
2
answers
1k
views
Are the real numbers the unique Dedekind-complete ordered set?
A totally ordered set is Dedekind-complete if any subset which has an upper bound also has a least upper bound. Now any two ordered fields which are Dedekind-complete are order-isomorphic as well as ...
- 10.3k
5
votes
2
answers
1k
views
How to define the operation of division apart from the inverse of multiplication?
Sorry if this question is too far out there, but I'm looking for a rigorous definition of the division operation. As I have seen it before, $a/b$ is the solution to the equation $a=xb$. While I am ...
- 2,049
4
votes
1
answer
183
views
Two uncountable subsets of real numbers without any interval and two relations
Are there two uncountable subsets $A, B$ of real numbers such that:
(1) $(A-A)\cap (B-B)=\{ 0\}$,
(2) $(A-A)+B=\mathbb{R}$ or $(B-B)+A=\mathbb{R}$ ?
We know that if one of them contains an interval,...
- 2,491
4
votes
1
answer
710
views
$n$-dimensional integer space? Or $\{ \mathbf{x} \in \mathbb{R}^n | x_1, x_2, ..., x_n \in \mathbb{Z} \}$?
If $\mathbf{x} \in \mathbb{R}^n$, then we would have $x_1, x_2, ..., x_n \in \mathbb{R}$, right? This is commonly known as $n$-dimensional space.
My question is, could we also have such a thing as $\...
- 4,312
2
votes
1
answer
195
views
Is this an isomorphism possible?
I am working on the following homework problem:
Let $\phi$ be an isomorphism from $\mathbb{R}^*$ to $\mathbb{R}^*$ (nonzero reals under multiplication). Show that if $r>0$, then $\phi(r) > 0$.
...
- 1,852
1
vote
2
answers
267
views
real numbers of the form $\frac{m}{10^n} $ with $m,n \in \mathbb{Z} $ and $n \geq 0$ is dense in $\mathbb{R}$ . [duplicate]
Problem :
Verify if the statement if true of false -
The set $S$ of all real numbers of the form $\frac{m}{10^n} $ with $m,n \in \mathbb{Z} $ and $n \geq 0$ is dense in $\mathbb{R}$ .
I think this ...
- 2,208
1
vote
3
answers
171
views
What are all different (non-isomorphic) field structures on $\mathbb R \times \mathbb R$
We know that $\mathbb R \times \mathbb R$ forms a field under addition and multiplication defined as $(a,b)+(c,d)=(a+c,b+d)$ ; $(a,b)*(c,d)=(ac-bd,ad+bc)$ ; is there any other way to make $\mathbb R \...
- 8,387
18
votes
3
answers
2k
views
Why is it so hard to prove a number is transcendental?
While reading on Wikipedia about transcendental numbers, i asked myself:
Why is it so hard and difficult to prove that $e +\pi, \pi - e, \pi e,
\frac{\pi}{e}$ etc. are transcendental numbers?
...
- 1,154
14
votes
2
answers
554
views
Can $\mathbb{R}^{+}$ be divided into two disjoint sets so that each set is closed under both addition and multiplication?
Can $\mathbb{R}^{+}$ be divided into two disjoint nonempty sets so that each set is closed under both addition and multiplication?
I know if we only require both sets to be closed under addition then ...
- 169
5
votes
3
answers
837
views
Is $\mathbb{Q}$ isomorphic to $\mathbb{Z^2}$?
Most of us are aware of the fact that $\mathbb{C}$ is isomorphic to $\mathbb{R^2}$, as we can define $\mathbb{C}$ as follows :
$$\mathbb{C} := \left\{z : z=x+iy \ \ \ \text{where} \ \ \langle x,y \...
- 13.2k