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 ...
- 940
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
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
12
votes
3
answers
460
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 ...
- 21.3k
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
262
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, ...
- 3,247
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 ...
- 90.8k
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 ...
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