All Questions
6
questions
0
votes
3
answers
443
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What is the smallest infinite field?
The real numbers and the rational numbers are both fields, but what is the smallest field. Is the set of rational numbers smaller than the set of reals, and if so is there a 'smaller' infinite set ...
1
vote
1
answer
91
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Alternative to the proof on Wolfram Mathworld that $\mathbb Q$ is the smallest subfield of $\mathbb R$
Exercise 2 (b) on page 100 of Analysis I by Amann and Escher asks me to show that $\mathbb Q$ is the smallest subfield of $\mathbb R$.
Wolfram MathWorld gives the following reasoning:
I don't know ...
- 4,252
6
votes
4
answers
207
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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
1
vote
1
answer
195
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Proof on Rational Numbers
I am trying to determine whether the following structure forms a Ring under the Real Number Definition of Addition and Multiplication
Consider the set of Real Numbers of the form:
$A = \{a + bp \:|\:...
user150203
5
votes
3
answers
837
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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
3
votes
1
answer
180
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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 ...
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