Let $K$ be an extencion of $F$ and let $a,b\in K$. If $a$ is not algebraic in $F$, but it is in $F(b)$, show that $b$ is algebraic in $F(a)$
118874
If $v$ is algebraic over $K(u)$, for some $u\in F$, and $v $ is transcendental over $K$, then $u$ is algebraic over $K(v)$
745
$F(x)$ versus $F[x]$ where $F$ is a field
55492
Existence of $f'\in F$ such that $f^{[G:F]}=(f')^n$ for finite extension $F\subset G$
88201
Suppose $X$ is a finite set and $f : X \to X$ is a function. Then $f$ is injective if and only if $f$ is surjective.
66946
show that $(i+k)! > i!k!$ algebraically
94190
Disproving the statement "If $f(z)$ is not an entire function, then $g(z) = f^2(z)$ is not entire.”
85977
Show that if $\sqrt{a}+\sqrt{b}\neq 0$, then $\mathbb{Q}(\sqrt{a}+\sqrt{b}) = \mathbb{Q}(\sqrt{a},\sqrt{b}), \forall a,b\in\mathbb{Q}$
158353
If $F=K(u,v)$ with $u^p$,$v^p\in K$ and $[F:K]=p^2$, $\operatorname{char} K=p>0$, then $F$ is not a simple extension of $K$.
79679
Let $\alpha, \beta \in \mathbb{C}$, such that $\alpha + \beta$, and $\alpha\beta$ are algebraic. Show that $\alpha$ and $\beta$ are algebraic.
125585
If $z=a$ is not the removable singularity of $f$, show that $e^{f(z)}$ has essential singularity at $z=a$.
121395
Let $X$ and $Y$ be non-empty sets and let $f:X \rightarrow Y$. Prove that $ f^{ −1} :P(Y ) → P(X) $ is injective if, and only if, $f$ is surjective.
71857
Proving that, $|f'(x)-f'(y)|\le k|x-y| \implies (f'(x))^2< 2 kf(x) $
79061
show that {$(x_1,x_2,x_3,x_4)\in\mathbb{F}^4$:$x_3=5x_4+b$} is a subspace
46056
If $\mathcal{E}$ are subsets of $X$ and $A \subset X$. Show that $\mathcal{A}\mathcal{(E \cap}$ A)=$\mathcal{A}\mathcal{(E)} \cap$ A
108633
Why $f^{-1}(f(A)) \not= A$
140095
Let $k^p = k$, $K = k(x)$ and $F = k(u)$ for some $u \in K$. Show that $K/F$ is separable $\iff u \not\in K^p$.
83714
$A \in B$ vs. $A \subset B$ for proofs