query: Proving $Gal(K/\mathbb{Q})$ is $D_4$.
positive_passages.docid: 22632
positive_passages.text: Galois group of $x^4-2$
negative_passages.docid: 11003
negative_passages.text: Proving that $G/N$ is an abelian group
negative_passages.docid: 99999
negative_passages.text: A question on identifying normal subgroups of $S_4$
negative_passages.docid: 4333
negative_passages.text: Prove that if $a+b+c+d=4$, then $(a^2+3)(b^2+3)(c^2+3)(d^2+3)\geq256$
negative_passages.docid: 37928
negative_passages.text: Isomorphism between $D_{2n}/Z(D_{2n})$ and $D_n$
negative_passages.docid: 42213
negative_passages.text: Computing $\text{Gal}(\mathbb{Q}^{ab}/\mathbb{Q})$
negative_passages.docid: 111291
negative_passages.text: Galois group of $\mathbb Q(\sqrt{4+\sqrt 7})/\mathbb Q$
negative_passages.docid: 128060
negative_passages.text: Proof that the relation $5 \mid (a + 4b)$ is symmetric and transitive
negative_passages.docid: 93029
negative_passages.text: Prove that when $d$ is in $\mathbb{Q} \setminus\mathbb{Z}$, $\mathbb{Z}[\sqrt{d}]$ is not a finitely generated abelian group.
negative_passages.docid: 138058
negative_passages.text: Prove that $\mathbb Z_{m}\times\mathbb Z_{n} \cong \mathbb Z_{mn}$ implies $\gcd(m,n)=1$.
negative_passages.docid: 52261
negative_passages.text: Proof that any self-complementary graph has to have $4k$ or $4k+1$ vertex, for some $k \in \mathbb N$
negative_passages.docid: 139451
negative_passages.text: Prove that any two nontrivial subgroups of $\mathbb{Q}$ have nontrivial intersection
negative_passages.docid: 1914
negative_passages.text: General way to determine $\mathbb{Q}(\gamma) = \mathbb{Q}(\alpha,\beta)$ given $\alpha$ and $\beta$
negative_passages.docid: 97258
negative_passages.text: Give $\ker(\phi)$ for $\phi: \mathbb{Z} \to \mathbb{Z_7}$ such that $\phi(1)=4$
negative_passages.docid: 4648
negative_passages.text: Proving that $\mathbb{Z}_m\oplus \mathbb{Z}_n \cong \mathbb{Z}_d\oplus \mathbb{Z}_l $ as groups, where $l=\mathrm{lcm}(m,n)$ and $d=\gcd(m,n)$
negative_passages.docid: 31451
negative_passages.text: Proving a subring of $\mathbb{Q}$ containing $\mathbb{Z}$ is a PID
negative_passages.docid: 38153
negative_passages.text: Looking for a simple proof that groups of order $2p$ are up to isomorphism $\mathbb{Z}_{2p}$ and $D_{p}$ .