query: Is $\mathbb{Z}_p \times \mathbb{Z}_p$ a field when $p$ is prime?
positive_passages.docid: 148192
positive_passages.text: In a field that is the direct product of two fields, why isn't it a field?
negative_passages.docid: 125568
negative_passages.text: Construction of $p^n$ field
negative_passages.docid: 154239
negative_passages.text: Can a prime in a Dedekind domain be contained in the union of the other prime ideals?
negative_passages.docid: 147200
negative_passages.text: Given fields $K\subseteq L$, why does $f,g$ relatively prime in $K[x]$ imply relatively prime in $L[x]$?
negative_passages.docid: 64068
negative_passages.text: Is every finite Galois extension of $\mathbb{Q}_p$ the completion of a number field at some valuation?
negative_passages.docid: 55429
negative_passages.text: Why does a Galois field have to have an order of $p^n$ where $p$ is prime?
negative_passages.docid: 94429
negative_passages.text: Alternative methods to prove that $Z_m$ is a field under addition and multiplication $\bmod\ m$ iff $m$ is a prime
negative_passages.docid: 37351
negative_passages.text: Group of Order $p^2$ Isomorphic to $\mathbb{Z}_{p^2}$ or $\mathbb{Z}_{p}$ $\times$ $\mathbb{Z}_{p}$
negative_passages.docid: 78342
negative_passages.text: When is $\phi(n)$ a prime number?
negative_passages.docid: 95937
negative_passages.text: Why aren't all prime subfields equal to {0,1}?
negative_passages.docid: 96549
negative_passages.text: Can we have a perfect field of characteristic $p$ with a non-archimedean valuation?
negative_passages.docid: 74663
negative_passages.text: Is $\mathbb{R}$ an algebraic extension of some proper subfield?
negative_passages.docid: 2235
negative_passages.text: Showing $\text{Aut}_\mathsf{Grp}(\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}) \cong S_3$
negative_passages.docid: 252
negative_passages.text: Why doesn't $0$ being a prime ideal in $\mathbb Z$ imply that $0$ is a prime number?
negative_passages.docid: 87780
negative_passages.text: Is the ideal $(2,X+1)\subset\Bbb{Z}[X]$ prime, maximal or neither?
negative_passages.docid: 99884
negative_passages.text: Are $\mathbb{Q}_p$ and $\mathbb{Q}_q$ homeomorphic?
negative_passages.docid: 128652
negative_passages.text: Subgroups of $\mathbb Z \times(\mathbb Z/n\mathbb Z)$