query: For non-algebraically closed field $k$ and integer $n>1$, there is a polynomial in $n$-variables over $k$ having only one zero in $k^n$
positive_passages.docid: 111338
positive_passages.text: Affine variety over a field which is not algebraically closed can be written as the zero set of a single polynomial
negative_passages.docid: 48873
negative_passages.text: Let $K$ be a field and $f(x)\in K[X]$ be a polynomial of degree $n$. And let $F$ be its splitting field. Show that $[F:K]$ divides $n!$.
negative_passages.docid: 127056
negative_passages.text: How to prove that a polynomial of degree $n$ has at most $n$ roots?
negative_passages.docid: 10885
negative_passages.text: Existence of irreducible polynomials over finite field
negative_passages.docid: 80002
negative_passages.text: zeros of a polynomial
negative_passages.docid: 138083
negative_passages.text: $\gcd(X^n - 1, nX^{n-1}) = 1$ in a field $K$ of characteristic $0$ or characteristic $p \nmid n$.
negative_passages.docid: 53669
negative_passages.text: Prove that the polynomial $(x-1)(x-2)\cdots(x-n) + 1$, $ n\ge1 $, $ n\ne4 $ is irreducible over $\mathbb Z$
negative_passages.docid: 46798
negative_passages.text: Is it possible that the zeroes of a polynomial form an infinite field?
negative_passages.docid: 70249
negative_passages.text: Irreducibility of Polynomials in $k[x,y]$
negative_passages.docid: 125604
negative_passages.text: Proof that $\mathbb{Z}$ has no zero divisors
negative_passages.docid: 6484
negative_passages.text: Polynomials in one variable with infinitely many roots.
negative_passages.docid: 67001
negative_passages.text: Factorizing polynomials over fields other than $\mathbb{C}$
negative_passages.docid: 111569
negative_passages.text: Property of a polynomial with no positive real roots
negative_passages.docid: 114146
negative_passages.text: Polynomial in $n$ variables
negative_passages.docid: 154373
negative_passages.text: How many irreducible monic quadratic polynomials are there in $\mathbb{F}_p[X]$?
negative_passages.docid: 4000
negative_passages.text: If every polynomial of degree 2 is reducible then $K$ is algebraically closed?
negative_passages.docid: 109721
negative_passages.text: The number of zeros in a polynomial