query: If $\alpha$ and $\beta$ are roots of monic polynomials in $\mathbb{Z}[x]$, is $\alpha + \beta$?
positive_passages.docid: 24551
positive_passages.text: How to prove that the sum and product of two algebraic numbers is algebraic?
negative_passages.docid: 154507
negative_passages.text: If all the roots of a polynomial P(z) have negative real parts, prove that all the roots of P'(z) also have negative real parts
negative_passages.docid: 111569
negative_passages.text: Property of a polynomial with no positive real roots
negative_passages.docid: 122759
negative_passages.text: Show $2+\alpha$ is a primitive root of $\mathbb{F}_{25}$.
negative_passages.docid: 120699
negative_passages.text: Is $\mathcal{O}_{\mathbb{Q}(\sqrt{5})} = \mathbb{Z}[\phi]$?
negative_passages.docid: 141124
negative_passages.text: Automorphisms of the ring $\Bbb Z[x]$ of polynomials with integer coefficients
negative_passages.docid: 131893
negative_passages.text: How to find the roots of $ x^4 + x + 1$ in the extension of $\Bbb F_2$?
negative_passages.docid: 99835
negative_passages.text: Polynomial in $\mathbb{Q}[x]$ with root $\sqrt[3]{2}+\sqrt[3]{3}$
negative_passages.docid: 102464
negative_passages.text: Prove $R < ( 1 + (\sum |a_{i}|^{p} )^{q/p})^{1/q}$ for absolute value of polynomial zeros
negative_passages.docid: 68790
negative_passages.text: Do we have $K(\beta)=K(\beta^2)$ for field extension of odd degree?
negative_passages.docid: 133312
negative_passages.text: Are these polynomials irreducible in $\mathbb{Q}[x]$?
negative_passages.docid: 68915
negative_passages.text: What are the units in $\mathbb{Z}[\root 3 \of 2]$?
negative_passages.docid: 137669
negative_passages.text: $P$ is a monic polynomial of degree $n$ , then which are correct?
negative_passages.docid: 63409
negative_passages.text: If $x^2+ax+b=0$ has a rational root, show that it is in fact an integer
negative_passages.docid: 145509
negative_passages.text: Prove the roots of $p(z)+z^n\bar{p}(\frac{1}{z})$ lie on the unit circle
negative_passages.docid: 42810
negative_passages.text: Is $(\mathbb{Q},+)$ isomorphic to $(\mathbb{Q}^n,+)$?
negative_passages.docid: 75349
negative_passages.text: Minimal polynomial of $\sqrt[3]{2} + \sqrt{3}$