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.
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?
...
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 $f: \mathbb{Z} \to \mathbb{Z}$ ...
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 ...
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 ...
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 $\...
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 ...
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 ...
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.
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 ...
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 ...
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, ...
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.
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 ...
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 ...
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 :
$$\mathbb{C} := \left\{z : z=x+iy \ \ \ \text{where} \ \ \langle x,y \...
5
votes
2
answers
1k
views
How to define the operation of division apart from the inverse of multiplication?
Sorry if this question is too far out there, but I'm looking for a rigorous definition of the division operation. As I have seen it before, $a/b$ is the solution to the equation $a=xb$. While I am ...
5
votes
1
answer
121
views
Is there a binary operation over the nonnegative reals satisfying the metric and group axioms?
Is there a binary operation over the nonnegative reals which satisfies the metric axioms and the group axioms? I.e., find an $f : S \times S \to S$ such that $(f,S)$ follows the group and metric ...
4
votes
1
answer
456
views
Is there any proper subring of $\mathbb{R}$ with field of fractions equal to $\mathbb{R}$? [duplicate]
Is there any proper subring of $\mathbb{R}$ with field of fractions equal to $\mathbb{R}$? Can we construct that proper subring? Is it necessarily an integral domain?
Updated: Is there an example to ...
4
votes
1
answer
183
views
Two uncountable subsets of real numbers without any interval and two relations
Are there two uncountable subsets $A, B$ of real numbers such that:
(1) $(A-A)\cap (B-B)=\{ 0\}$,
(2) $(A-A)+B=\mathbb{R}$ or $(B-B)+A=\mathbb{R}$ ?
We know that if one of them contains an interval,...
4
votes
1
answer
3k
views
Existence of identity element for binary operation on the real numbers.
We define a new operation
$$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 ...
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{...
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 ...
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 $\...
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)$
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}$. ...
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 ...
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, ...
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 ...
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)...
3
votes
1
answer
136
views
Is there an ordered field with distinct subfields isomorphic to the reals?
Is there an ordered field with distinct subfields isomorphic to the field $\mathbb R$ of real numbers?
3
votes
1
answer
87
views
Historically, when have the the real numbers been constructed via the "positive" (non-negative) reals first, and then usual real numbers second?
There has been something that has been bugging me for the longest time, at least since grad school.
In the teaching of mathematics, during the construction of the "usual" real numbers, why ...
3
votes
1
answer
243
views
Find all $\mathbb{Q}$ subspaces of $\mathbb{Q} \times \mathbb{R}$
While working to Apply Lam's theorem to determine all the left ideals of $\begin{pmatrix}\mathbb{Q}&\mathbb{R}\\0&\mathbb{R}\end{pmatrix}$ I have encountered the problem of determining all ...
3
votes
1
answer
180
views
Dedekind's Cuts Lemma
I'm studying Dedekind's Cuts and his construction of Real numbers from the Rational ones. Here we are allowed to use $\Bbb{Q}$ as an ordered field and all all its properties (Archimedean Property, his ...
3
votes
0
answers
112
views
Proof that for all nonzero real numbers $a$, $\frac{1}{a}$ is nonzero
I was wondering if someone could check my proof that "For all $a\in\mathbb{R}$, if $a\neq 0$ then $\frac{1}{a}\neq 0$". The definitions/assumptions I am basing the proof off of come from &...
2
votes
3
answers
245
views
Inverse of isomorphism between two fields is isomorphism
I'm using this definition of isomorphism between two systems satisfying the axioms of Dedekind-complete totally ordered fields to show that its inverse is an isomorphism too. I can be easily proved ...
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$.
...
2
votes
2
answers
144
views
Is there a reasonable limit to how far you can generalise complex numbers? [duplicate]
Real numbers satisfy a(bc) = (ab)c as well as ab = ba. They are also comparable.
Generalising to complex numbers, everything stays the same, except the numbers lose their comparibility.
Generalising ...
2
votes
1
answer
635
views
Is an automorphism of the field of real numbers without ordering the identity map? [duplicate]
We know an automorphism on $\mathbb{R}$ must fix $\mathbb{Q}$. If we assume the usual order structure and topology on $\mathbb{R}$, then we can use the density of $\mathbb{Q}$ to show an automorphism ...
2
votes
1
answer
129
views
If $a,b$ and $b,c$ are algebraic, can we effectively say $a,c$ are algebraic?
Lets say for two transcendental complex numbers x,y in $\Bbb C$ that they are algebraically independent, if there doesn't exists a non-zero polynomial $p(X,Y) \in \Bbb Q[X,Y]$, such that:
$$p(x,y)=0$$...
2
votes
1
answer
99
views
If A is a square matrix of size n with real entries, with $A = A^{p+1}$, then $rank(A) + rank (I_n - A^p) = n$
Let A be a square matrix of size n with real entries, $n \geq 2$, with $A = A^{p+1}$, $p \geq 2 $, then $$rank(A) + rank (I_n - A^p) = n$$
If p is prime, in addition, $$rank (I_n - A)=rank (I_n - A^...
2
votes
1
answer
75
views
$(H,*)$ group with some properties $\Rightarrow H$ not an interval.
Let $(H, \ast)$ be a group, where $H \subseteq (0, \infty)$, which has these properties:
$x \in H \Rightarrow \frac{1}{x} \in H$
$2023 \in H$, and
$x \ast y = \frac{1}{x} \ast \frac{1}{y}$ for any $x,...
2
votes
1
answer
51
views
Can we define a new multiplication on $\Bbb R$ ( addition and identity element remain the same) while its properties of being a field persist? [closed]
If the identity element of $\Bbb R$ were permitted to be redefined, then it could be redefined as any nonzero number while keeping $\Bbb R$ a field;
By induction, we can prove that only the product ...
2
votes
2
answers
187
views
Homomorphism Between a Geometric Algebra and its Field of Scalars?
Given a geometric algebra defined over a real vector space, is is possible to construct a homomorphism between the elements of the geometric algebra and the reals?
I was pondering an example of this: ...
2
votes
1
answer
449
views
Question on the archimedean property
Let $a,b \in \Bbb ℝ$. Suppose that $a>0$. Prove that there is some $n\in \Bbb N$ such that $b\in[-na, na]$.
I understand how the Archimedean Property can be used to prove this statement if $b$ is ...
2
votes
1
answer
213
views
What would be interesting maps to use on that Eudoxus reals?
I'm trying to understand Eudoxus Reals. From wikipedia:
Let an almost homomorphism be a map $f:\mathbb{Z}\to\mathbb{Z}$ such that the set $\{f(n+m)-f(m)-f(n): n,m\in\mathbb{Z}\}$ is finite. We say ...
2
votes
0
answers
88
views
Finding equivalent submodular function
Let $V$ be a finite set of points in $\mathbb{R}^n$ with $d(x,y)$ denoting the usual Euclidean distance between two points $x$ and $y$. I am trying to order points from the set $V$ iteratively, at ...
2
votes
0
answers
48
views
Basic proof that every polynomial over $\mathbb R$ factorizes into at most quadratic ones
Is there a proof without using that in $\mathbb C$ every polynomial factorizes into linear ones that every polynomial over $\mathbb R$ factorizes into linear or quadratic ones?
2
votes
0
answers
133
views
Extended real numbers as algebraic structure
I need to work with real numbers, but extended to have an additional element. This element, I denote by $\odot$ and my set is: $\mathbb{R}_{\odot}=\mathbb{R}\cup\{\odot\}$. This element should behave ...
2
votes
2
answers
3k
views
Prove that $\sup f(x) \leq \inf g(y)$
Let $f: D \longrightarrow \mathbb{R}$ and $g: D\longrightarrow \mathbb{R}$ be functions ($D$ nonempty). Suppose that $f(x) \leq g(y)$ for all $x\in D$ and $y \in D$. Show that
$$\sup f(x) \leq \inf g(...