Intereting Posts

book for metric spaces
Definition of Basis for the Neighborhood System
Are $121$ and $400$ the only perfect squares of the form $\sum\limits_{k=0}^{n}p^k$?
Why are convex polyhedral cones closed?
The fibers of a finite morphism of affine varieties are all finite
Solution of $A^\top M A=M$ for all $M$ positive-definite
Revisted$_2$: Are doubling and squaring well-defined on $\mathbb{R}/\mathbb{Z}$?
Why aren't logarithms defined for negative $x$?
showing that the sequence $a_n=1+\frac{1}{2}+…+\frac{1}{n} – \log(n)$ converges
Why is a straight line the shortest distance between two points?
How would you explain why “e” is important? (And when it applies?)
Groups in an abstract algebra
Show that any bounded linear functional on a normed linear space is continuous
Dimension of the solution of a second order homogenous ODE
$\sin 1^\circ$ is irrational but how do I prove it in a slick way? And $\tan(1^\circ)$ is …

I’m trying to determine if there is a field, F, such that $\mathbb{R}$ $\subsetneq$ F $\subsetneq$ $\mathbb{C}$ where F is not the same as $\mathbb{R}$ or $\mathbb{C}$.

- Dimension of an algebraic closure as a vector space over its base field.
- Find primitive element such that conductor is relatively prime to an ideal (exercise from Neukirch)
- Find a non-principal ideal (if one exists) in $\mathbb Z$ and $\mathbb Q$
- Minimum polynomial of $\sqrt{2} + \sqrt{5}$ above $\mathbb{Q}$ (and a generalization)
- Roots in different algebraic closure have the same multiplicative relations
- Minimal polynomial of intermediate extensions under Galois extensions.
- Examples of fields of characteristic 0?
- Is there a proper subfield $K\subset \mathbb R$ such that $$ is finite?
- If $(F:E)<\infty$, is it always true that $\operatorname{Aut}(F/E)\leq(F:E)?$
- Is $\mathbb{Q}(\sqrt{3}, \sqrt{3})$ a Galois extension of $\mathbb{Q}$

There is not. The extension $[\mathbb C : \mathbb R]$ has degree 2, so there cannot be a proper intermediate extension.

Assume such an extension $F$ exists. Then by the tower theorem, we have $[\mathbb C:F][F:\mathbb R]=2$. For it to be a proper extension, each factor must be greater than one. This is a contradiction.

Suppose $F$ is a field such that $\mathbb{R} \subsetneq F \subseteq \mathbb{C}.$ Then $F$ contains at least one element $a+bi$ where $b\neq 0.$ Since $F$ contains all the reals, it contains $b^{-1}$ and $-a,$ so it contains $b^{-1}\left((a+bi)-a\right)=i.$ Hence it contains all complex numbers and $F=\mathbb{C}.$

- Show that $A\in\mathbb{C}_n$ is normal $\iff$ $tr(A*A) = \sum_{i = 1}^n|\lambda_i|^2$, where $\lambda_1,…,\lambda_n$ are the eigenvalues of $A$.
- Trying to understand a “Scott open set” for the “Scott Topology”
- What are the practical applications of this trigonometric identity?
- A problem on field extension
- Inverse of a Positive Definite
- Number of cycles in complete graph
- Turing's 1939 paper on ordinal logic
- Every sentence in propositional logic can be written in Conjunctive Normal Form
- Let $T,S$ be linear transformations, $T:\mathbb R^4 \rightarrow \mathbb R^4$, such that $T^3+3T^2=4I, S=T^4+3T^3-4I$. Comment on S.
- Intuitive Explanation of Bessel's Correction
- Do linear continua contain $\mathbb{R}$? Can a nontrivial connected space have only trivial path components?
- Axiom of Regularity
- Is there a topological space which is star compact but not star countable?
- What is your definition for neighborhood in topology?
- How to prove a total order has a unique minimal element