All Questions
Tagged with real-numbers abstract-algebra
98
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
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
16
votes
1
answer
168
views
What is the "higher cohomology" version of the Eudoxus reals?
The "Eudoxus reals" are one way to construct $\mathbb{R}$ directly from the integers. A full account is given by Arthan; here is the short version: A function $f: \mathbb{Z} \to \mathbb{Z}$ ...
- 1,247
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 ...
- 950
14
votes
2
answers
556
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
13
votes
2
answers
533
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
12
votes
3
answers
462
views
Is there a minimal generating set of reals which additively generate all the reals?
Is there a set $S$ of real numbers such that the submagma generated by $S$ under addition is the entire set of real numbers, but such that no proper subset of $S$ generates the entire set of real ...
- 21.5k
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
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
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
8
votes
1
answer
1k
views
Is this a field?
Let $S$ be the set of all the ordered pairs in the cartesian plane. That is:
$$S=\{(x,y)|\ \ x, y \in \Bbb{R}\}$$
Then, If $a=(a_1, a_2)$ and $b=(b_1, b_2)$ are two arbitrary elements of $S$, the ...
- 1,598
8
votes
1
answer
263
views
$A,B$ such that $A\cap B=\emptyset$ and $A\cup B=\mathbb{R}$ and $B=\{x+y : x,y\in A\}$?
If set $A,B$ satisfy $A\cap B=\emptyset,A\cup B=I$, and $B=\{x+y : x,y\in A\}$, can $I$ be real number set $\mathbb{R}$?
I think the answer is yes, but I can't construct it. If $A$ is odd number set, ...
- 3,267
6
votes
4
answers
207
views
Is $\mathbb Q$ a quotient of $\mathbb R[X]$?
Is there some ideal $I \subseteq \mathbb R[X]$ such that $\mathbb R[X]/I \cong \mathbb Q$?
$I$ is clearly not a principal ideal.
- 10.5k
6
votes
2
answers
392
views
Path From Positive Dedekind Cuts to Reals?
Don't spend a lot of time on this. I'm certain I could bang it out myself; but maybe there's an answer out there that someone already knows.
Say we use Dedekind cuts to construct the reals. Addition ...
- 90.9k
6
votes
2
answers
118
views
Two "different" definitions of $\sqrt{2}$
In Walter Rudin's Principles of Mathematical Analysis (3rd edition) (page 10), it is proved that
for every $x>0$ and every integer $n>0$ there is one and only one positive real $y$ such that ...
user765539
5
votes
3
answers
843
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 :
$$\mathbb{C} := \left\{z : z=x+iy \ \ \ \text{where} \ \ \langle x,y \...
- 13.2k
5
votes
2
answers
1k
views
How to define the operation of division apart from the inverse of multiplication?
Sorry if this question is too far out there, but I'm looking for a rigorous definition of the division operation. As I have seen it before, $a/b$ is the solution to the equation $a=xb$. While I am ...
- 2,049
5
votes
1
answer
121
views
Is there a binary operation over the nonnegative reals satisfying the metric and group axioms?
Is there a binary operation over the nonnegative reals which satisfies the metric axioms and the group axioms? I.e., find an $f : S \times S \to S$ such that $(f,S)$ follows the group and metric ...
user275490
4
votes
1
answer
456
views
Is there any proper subring of $\mathbb{R}$ with field of fractions equal to $\mathbb{R}$? [duplicate]
Is there any proper subring of $\mathbb{R}$ with field of fractions equal to $\mathbb{R}$? Can we construct that proper subring? Is it necessarily an integral domain?
Updated: Is there an example to ...
- 420
4
votes
1
answer
183
views
Two uncountable subsets of real numbers without any interval and two relations
Are there two uncountable subsets $A, B$ of real numbers such that:
(1) $(A-A)\cap (B-B)=\{ 0\}$,
(2) $(A-A)+B=\mathbb{R}$ or $(B-B)+A=\mathbb{R}$ ?
We know that if one of them contains an interval,...
- 2,491
4
votes
1
answer
3k
views
Existence of identity element for binary operation on the real numbers.
We define a new operation
$$x*y= x+y+xy,$$
on the set of real numbers with the usual addition and multiplicaton. Has this operation got an identity element?
It seems clear for me that there is one ...
- 203
4
votes
1
answer
109
views
Elements of $\operatorname{Hom}_{\mathbb{Z}} (\mathbb{Q}, \mathbb{Q}/\mathbb{Z}) $ as Cauchy sequences
There is an isomorphism of abelian groups $\operatorname{Hom}_{\mathbb{Z}} (\mathbb{Q}, \mathbb{Q}/\mathbb{Z}) \cong \mathbb{R}$, a proof is based on three observations:
$\operatorname{Hom}_{\mathbb{...
- 6,407
4
votes
1
answer
177
views
Abstract concept tying real numbers to elementary functions?
Real numbers can be broken into two categories: rational vs. irrational. Irrational numbers can be approximated, but never fully represented by rational numbers.
Analytic functions have Taylor ...
- 247
4
votes
1
answer
715
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 $\...
- 4,322
3
votes
1
answer
487
views
Homomorphism from $\mathbb R^2\to \mathbb C$
Is it possible to define a surjective ring homomorphism from $\mathbb R^2$ onto $\mathbb C$? The multiplication defined on $\mathbb R^2$ is as follows:
$(a,b)(c,d)=(ac,bd)$
- 4,928
3
votes
1
answer
435
views
Why is $\mathbb{R}/\mathbb{Z}$ not an $\mathbb{R}$-vector space?
This is an embarrassing question which might seem elementary and possibly silly, but its suddenly confusing me. Clearly I'm missing something very obvious.
Take the structure $\mathbb{R}/\mathbb{Z}$. ...
- 2,517
3
votes
2
answers
923
views
Why is the observation that proof of the Fundamental Theorem of Algebra requires some topology not tautological?
I have heard it mentioned as an interesting fact that the Fundamental Theorem of Algebra cannot be proven without some results from topology. I think I first heard this in my middle school math ...
- 1,387
3
votes
1
answer
95
views
Why there would be no additive inverse in real numbers if we changed the definition of a cut?
We define a cut to be a proper subset of rationals such that:
1- It is not the empty set $\emptyset$,
2- It is closed to the left, meaning that if $p \in \alpha, q<p \Rightarrow q \in \alpha.$
So, ...
- 138
3
votes
1
answer
76
views
Sets of real numbers which are anti-closed under addition
Let $(M,*)$ be a magma, that is, a set with a binary operation. I define a subset $S$ of $M$ to be anti-closed under $*$ iff for all $x,y$ in $S$, $x*y \notin S$. For example, the set of negative real ...
- 21.5k
3
votes
1
answer
206
views
Abstracting Magnitude Measurement Systems (i.e. subsets of ${\mathbb R}^{\ge 0}$) via Archimedean Semirings.
I did some googling but could not find any easily accessible theory so I am going to lay out my ideas and ask if they hold water.
Definition: A PM-Semiring $M$ satisfies the following six axioms:
(1)...
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