All Questions
Tagged with real-numbers abstract-algebra
98
questions
1
vote
1
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131
views
Which algebraic intuition can be used in fields
I wonder what basic laws of arithmetic of reals e.g. $x^n x^m = x^{m+n}$ holds for fields. Every time I take some book on abstract algebra it proves very abstract and unpractical properties. So I ...
- 2,420
1
vote
1
answer
330
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What properties do you lose when you extend your number set? [duplicate]
So in $\mathbb{R}$ and $\mathbb{C}$ you have both associative and commutative property, but as you extend to $\mathbb{H}$ you lose the commutative property, and $\mathbb{O}$ loses the associativity. ...
- 5,339
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 ...
- 2,811
1
vote
3
answers
171
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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
2
votes
2
answers
3k
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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(...
- 349
0
votes
1
answer
70
views
Related Zorn's lemma proof?
Let $S$ be a partially ordered set, with the additional property that every chain $s_0\le s_1 \le s_2 \le...$ has an upper bound in $S$ (i.e. there is some $t$ in $S$ such that $s_n \le t$ for all $n$)...
- 27
76
votes
7
answers
33k
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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.
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