$f$ has a root in $F$ iff $f$ has a root in $\text{Frac}(F)$
85782
UFDs are integrally closed
72694
Proving that $f(x) = g(x)$ for some $x \in [a,b]$ if $f,g$ continuous, $f(a)  g(b)$
2072
Demonstrate that if $f$ is surjective then $X = f(f^{-1}(X))$
250
An irreducible $f\in \mathbb{Z}[x]$, whose image in every $(\mathbb{Z}/p\mathbb{Z})[x]$ has a root?
115299
Elements of $\mathbb{F}_p$ having cube roots in $\mathbb{F}_p$
154363
Prove that $\text{aff}(X) = \text{aff}(\text{closure}(X))$
98474
Show the existence of a certain subgroup of F2
118266
If $f^2(b) - f^2(a) = b^2 - a^2$, then the equation $f'(x)f(x) = x$ has at least one root in (a, b).
18950
Show that $f$ has at most one fixed point
53024
$f(x)f(1/x)=f(x)+f(1/x)$
88278
Show $S = f^{-1}(f(S))$ for all subsets $S$ iff $f$ is injective
7858
Proving that $f(x)$ is irreducible over $F(b)$ if and only if $g(x)$ is irreducible over $F(a)$
50012
$f: [a,b] \to \mathbb{R}$, $f(a) = f(b)$. Show $\exists c \in [a, \frac{a+b}{2} ]$ with $f(c)=f(c+\frac{b-a}{2})$
45710
Finishing a proof: $f$ is injective if and only if it has a left inverse
17310
Can $f(x)=|x|^3$ be differentiated
4706
Prove $f=x^p-a$ either irreducible or has a root. (arbitrary characteristic) (without using the field norm)
6722
to show $f$ is a polynomial in $z$