Questions tagged [abstract-algebra]
For questions about monoids, groups, rings, modules, fields, vector spaces, algebras over fields, various types of lattices, and other such algebraic objects. Associate with related tags like [group-theory], [ring-theory], [modules], etc. as necessary to clarify which topic of abstract algebra is most related to your question and help other users when searching.
87,910 questions
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Inflection point on elliptic curves
I've been having trouble coming up with a full-fleged answer for a question in a homework assignment for my algebraic geometry class. The question says:
Let $F$ be an elliptic curve (projective cubic ...
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classification of f.g. abelian groups isomorphism
I am trying to solve the following problem:
A partition of n is a sequence of positive integers $(k_1,\dots,k_m)$ such that $k_1≥k_2≥\dots$ and $\sum k_i=n$.
Prove that there is a bijection between ...
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Localization at a prime of a subring preserves noetherianity
Let $B \subseteq A$ be an extension of noetherian rings, and let $\mathfrak{p} \trianglelefteq B$ be a prime ideal. The localization of $A$ (viewed as a $B$-module) at $\mathfrak{p}$ is $A \otimes_B ...
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Are there existing results on the number of solutions to $x^2 - y^2 = n$ in the quadratic ring $\mathbb{Z}_{m}[\sqrt D]$?
Let $n \in \mathbb{Z}$. I am trying to work on the equation $x^2 - y^2 = n$ in the ring of quadratic integers modulo $m$, defined as $\mathbb{Z}_m[\sqrt D] = \{a+b\sqrt D \mid a,b \in \mathbb{Z}_m\}$, ...
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Robinson's word vs Rotman's word: can the second be derived from the first?
In "A Course in the Theory of Groups" written by Derek J.S. Robinson it is done as follows.
Let be $X$ a set. Choose a set disjoint from $X$ with the same cardinality (see here for details):...
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For an adapted basis $\{ a_1y_1, \dots , a_ny_n\} \subseteq N \subseteq M$, $M/N \cong R/a_1R \times \cdots \times R/a_n R$ ? (And General Question)
Let $R$ be a Principal ideal domain, let $M$ be a free $R$-module of finite rank $n$ and let $N$ be a submodule of $M$ with same rank $n$. Then by the Dummit, Foote's Abstract Algebra, Chapter 12, ...
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Why are left and right cosets different in the non-normal case?
I understand that the only difference between left and right group actions is the order in which gh acts on x. Since cosets Hg and gH involve the action of one element of the subgroup H on one element ...
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How are left/right multiplication different from other group actions?
Thus, for establishing general properties of group actions, it suffices to consider only left actions. However, there are cases where this is not possible. For example, the multiplication of a group ...
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Prove that the minimal polynomial of $\frac{\sin{g \frac{\pi}{p}}}{\sin{\frac{\pi}{p}}}$ has degree $\frac{p-1}{2}$ over $\mathbb{Q}$.
Prove that the minimal polynomial of $\frac{\sin{g \theta}}{\sin{\theta}}$ has degree $\frac{p-1}{2}$, where $\theta=\frac{\pi}{p}$, $g$ is a primitive root $\mathrm{mod}$ $p$ and $2 \nmid g$.
My ...
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Conjecture: a constant-free equation is solvable in a group $G$ if and only if it is solvable in its generating set $B$ for any $G=\langle B\rangle$
Conjecture.
Let $G$ be a group and $B$ any set of generators for $G$. That is to say $G = (G, \cdot) = \langle B \rangle$. Then for any equation $E=F$ in $G$ that is constant-free, we have that $E=...
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inverse image of ideals in a ring of fractions (localization)
Let $R$ be a commutative ring (possibly without unity), and $S$ a multiplicative subset of $R$, and $\phi_S$ be the map $r \mapsto rs/s$ from $R$ to the ring of quotients $S^{-1}R$. If $I$ is an ideal ...
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If $H$ and $N$ are respectively a pronormal and normal subgroup then is $H\cap N$ pronormal in $N$?
If $H$ is a subgroup of a group $(G,\ast,e)$ then it is said pronormal iff for any $g$ in $G$ there exists $x$ in $\left\langle H\cup(g\ast H\ast g^{-1})\right\rangle$ such that the equality
$$
g\ast ...
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Grouping of Polynomial Ring Calculation [duplicate]
I am currently working my way through The Beginner’s Textbook
for Fully Homomorphic Encryption by
Ronny Ko. I cant wrap my head around how he grouped those terms up (p.100). If anybody could help me ...
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Attempting to prove $\text{Ext}_{\mathbb{Z}}^{2}(A, B)=0$ directly.
It's well known that $\text{Ext}_{\mathbb{Z}}^{n}(A, B)=0$ for $n\geq 2$ as a consequence of either projective/injective dimension. One can also view $\text{Ext}^{n}_{\mathbb{Z}}(A, B)$ as the group ...
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Showing $f(x)=2\cos 3x+3\sin x$ is not algebraic [closed]
Let the function $f$ be defined on $\mathbb{R}$ by $f(x)=2\cos 3x +
3\sin x. $
A function $f$ is called algebraic over $\mathbb{R}$ if there
exist polynomials $P_0(x),P_1(x),\dots,P_n(x)\in\mathbb{R[...
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