In which ways does Hilbert's Nullstellensatz fail if the field is not algebraically closed?
113742
Nullstellensatz non-valid for non-algebraically closed fields
157420
Question about Subrings and integraly closed
44502
An irreducible polynomial cannot share a root with a polynomial without dividing it
1720
Show that $\langle 2,x \rangle$ is not a principal ideal in $\mathbb Z [x]$
142336
Why does the integral domain "being trapped between a finite field extension" implies that it is a field?
19511
Is the sub-field of algebraic elements of a field extension of $K$ containing roots of polynomials over $K$ algebraically closed?
24247
Ring Inside an Algebraic Field Extension
69398
Rank-nullity theorem for free $\mathbb Z$-modules
116916
What's the rationale for requiring that a field be a $\boldsymbol{non}$-$\boldsymbol{trivial}$ ring?
19754
$\mathbb{R}$ \ $\mathbb{Q}$ and $\mathbb{R}^2\setminus\mathbb{Q}^2$ disconnected?
95594
Prove that a polynomial is irreducible or the field contains a $p$th root
139463
Commutative integral domain does not finitely generate its field of fractions
63338
Noncommutative failure of IBN: $\,R^n\cong R^m \Rightarrow n - m,$ ring $R$ with maximal ideal $I$ such that $R/I$ is not a division ring?
106737
A problem on field extension
154360
Show $f$ can't be irreducible over a finite field if $f^\prime$ is the zero polynomial.
121076
Show $x^p-t$ has no root in the field $\mathbb{F}_p(t)$
55864
Is there a null sequence that is in not in $\ell_p$ for any $p<\infty$?