Abstract algebra is the study of algebraic objects. Some of the more common algebraic objects are groups, rings, fields, vector spaces, modules, among other topics.

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What is $\operatorname{Hom}_R(P,R)$ isomorphic to when $P$ is projective?

Let $R$ be a (possibly noncommutative) ring with $1$. Now, quite clearly we have $$\operatorname{Hom}_R(R^n,R)\cong R^n.$$ I am wondering if there is any similar result for ...
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Showing all Frobenius forms are given by multiplication with an invertible element

Let $A$ be an algebra over a field $\mathbb{k}$. A Frobenius form is a linear functional $\varepsilon : A \rightarrow \mathbb{k}$ that contains no nontrivial left ideals in its nullspace. My ...
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Unsolvability of a Quintic and its link with “Simplicity” of $A_{5}$

At the outset I must mention that I don't have a fairly working knowledge of Galois Theory (but do have some idea of group theory in the sense that I can understand normal subgroups). I read the ...
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Meromorphic functions on $U^2 = T^3 + 1$, genus.

Let $k$ be a field of characteristic $\neq 2$, and consider the quadratic extension $F$ of $k(T)$ generated by $\sqrt{T^3 + 1}$. What is/how do I find the genus of $F$? The progress I have so far: ...