query: For every $n$, there exists a polynomial of degree $n$ with Galois Group $S_n$
positive_passages.docid: 82328
positive_passages.text: Constructing a Galois extension field with Galois group $S_n$
negative_passages.docid: 119999
negative_passages.text: A property of every real polynomial.
negative_passages.docid: 151729
negative_passages.text: Existence of a unique polynomial
negative_passages.docid: 38419
negative_passages.text: Existence of Polynomial with whom product contains only prime powers
negative_passages.docid: 127547
negative_passages.text: Prove that If $f$ is polynomial function of even degree $n$ with always $f\geq0$ then $f+f'+f''+\cdots+f^{(n)}\geq 0$.
negative_passages.docid: 8164
negative_passages.text: Product of all monic irreducibles with degree dividing $n$ in $\mathbb{F}_{q}$?
negative_passages.docid: 103246
negative_passages.text: Prove that a polynomial of degree $d$ has at most $d$ roots (without induction)
negative_passages.docid: 96680
negative_passages.text: Generators of symmetric group $S_n$
negative_passages.docid: 116856
negative_passages.text: Proving that a polynomial of the form $(x-a_1)\cdots(x-a_n) + 1$ is irreducible over $\mathbb{Q}$
negative_passages.docid: 13981
negative_passages.text: Proof that every polynomial of odd degree has one real root
negative_passages.docid: 99630
negative_passages.text: For every $g\in G$ there exists an $h\in G$ such that $g = h^3$
negative_passages.docid: 14890
negative_passages.text: Let $G=\langle a\rangle$ be a cyclic group of order $n$. Show that, for every divisor $d$ of $n$, there exists a subgroup of $G$ whose order is $d$.
negative_passages.docid: 127056
negative_passages.text: How to prove that a polynomial of degree $n$ has at most $n$ roots?
negative_passages.docid: 146312
negative_passages.text: Is the set of roots with respect to their values and multiplicies unique for each $g(x)$ in $a_n x^n + g(x)$?
negative_passages.docid: 147158
negative_passages.text: Galois group of a non-separable polynomial
negative_passages.docid: 56439
negative_passages.text: $f \in K[X]$ of degree $n$ with Galois group $S_n$, why are there no non-trivial intermediate fields of $K \subset K(a)$ with $a$ a root of $f$?
negative_passages.docid: 73669
negative_passages.text: Totally real Galois extension of given degree