query: Group isomorphism between $(\mathbb{Z}[x], +)$ and $(\mathbb{Q}^+, \cdot)$
positive_passages.docid: 157915
positive_passages.text: Isomorphism between the group $(\mathbb Z[x], +)$ and $(\mathbb Q_{>0}, .)$
negative_passages.docid: 141124
negative_passages.text: Automorphisms of the ring $\Bbb Z[x]$ of polynomials with integer coefficients
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: 126513
negative_passages.text: Automorphism of the group of rational numbers under addition
negative_passages.docid: 61732
negative_passages.text: Prove any group with 3 elements is isomorphic to $\mathbb{Z}_3$
negative_passages.docid: 45627
negative_passages.text: Find a sub-ring of M(2, $\mathbb{Q}$) which is isomorphic to $\mathbb{Q}$
negative_passages.docid: 77903
negative_passages.text: Finding the integral closure of $\mathbb{Z}$ in $\mathbb{Q}[i]$
negative_passages.docid: 42810
negative_passages.text: Is $(\mathbb{Q},+)$ isomorphic to $(\mathbb{Q}^n,+)$?
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: 59995
negative_passages.text: Isomorphism between $\mathbb{C}^* $ under multiplication with $\mathbb{C}$ under addition.
negative_passages.docid: 99032
negative_passages.text: Prove that the additive group of $\mathbb{Z}[x]$ is isomorphic to the group of positive rational numbers under multiplication.
negative_passages.docid: 2153
negative_passages.text: How to get a homomorphism from $\mathbb{Z}\to G$, for $G$ a non-trivial group?
negative_passages.docid: 157496
negative_passages.text: Determine the Galois group of $\mathbb{Q}(\sqrt{a+b\sqrt{d}})$
negative_passages.docid: 95602
negative_passages.text: Computing the Galois group of $x^4+ax^2+b \in \mathbb{Q}[x] $
negative_passages.docid: 17978
negative_passages.text: Isomorphism classes of $\mathbb{Z}[i]$ modules.
negative_passages.docid: 3203
negative_passages.text: Quotient group $\mathbb Z^n/\ \text{im}(A)$
negative_passages.docid: 72478
negative_passages.text: What is the isomorphism between the fields $(Z_2[x]^{<3},+_{x^3+x^2+1},\times_{x^3+x^2+1})$ and $(Z_2[x]^{<3},+_{x^3+x+1},\times_{x^3+x+1})$?