query: Why isn't $1 \to \mathbb{Z}_2 \to \mathbb{Z}_4 \to \mathbb{Z}_2 \to 1$ a split extension?
positive_passages.docid: 151830
positive_passages.text: Exact sequence splits?
negative_passages.docid: 26749
negative_passages.text: $\mathbb{Z}[X]/(X^2-1)$ is not isomorphic with $\mathbb{Z}\times \mathbb{Z}$
negative_passages.docid: 124864
negative_passages.text: A not free $\mathbb{Z}$-module
negative_passages.docid: 53335
negative_passages.text: Why isn't $\mathbb{C}[x,y,z]/(xz-y)$ a flat $\mathbb{C}[x,y]$-module
negative_passages.docid: 122312
negative_passages.text: How can I compute $\mathrm{Aut}(\mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/6\mathbb{Z})$?
negative_passages.docid: 40400
negative_passages.text: Finding an isomorphism between $\mathbb{Z}[i]/(7)$ and $\mathbb{Z}[\sqrt{-2}]/(7)$
negative_passages.docid: 146990
negative_passages.text: How to find the number of automorphisms on $\mathbb Z_2\times \mathbb Z_4$
negative_passages.docid: 66313
negative_passages.text: $\mathbb{Z}[i]/(a+bi) \cong \mathbb{Z}_{a^2+b^2}$ if $(a,b)=1$
negative_passages.docid: 20218
negative_passages.text: Is this field extension finite?
negative_passages.docid: 98587
negative_passages.text: $\mathbb Z [X] / (X)$ isomorphic to $\mathbb Z[X] / (X+1)$ isomorphic to $\mathbb Z [X] / (X+2015)$
negative_passages.docid: 80202
negative_passages.text: Why is $[0, 1] \cap \mathbb{Q}$ not compact in $\mathbb{Q}$?
negative_passages.docid: 78222
negative_passages.text: Can any monomorphism of a subfield of a splitting field be extended to an automorphism?
negative_passages.docid: 45991
negative_passages.text: Why $f\colon \mathbb{Z}_n^\times \to \mathbb{Z}_m^\times$ is surjective?
negative_passages.docid: 33025
negative_passages.text: What is the order of $(\mathbb{Z} \oplus \mathbb{Z})/ \langle (2,2) \rangle$ and is it cyclic?
negative_passages.docid: 85032
negative_passages.text: $Q(\mathbb{Z}[t]) / \mathbb{Z}[t]$ is not injective
negative_passages.docid: 24168
negative_passages.text: A characterisation of quadratic extensions contained in cyclic extensions of degree 4
negative_passages.docid: 33781
negative_passages.text: Is $\mathbb{Z}[x]/(x^2+1)$ isomorphic to $\mathbb{Z}[i]$?