# Tagged Questions

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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### Extending our number system to include infinities [closed]

Instead of writing infinity using the infinity symbol, could we write such numbers as: |$\mathbb Z$| (size of the set of integer numbers) |$\mathbb R$| (size of the set of real numbers) Then ...
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### Separable but not reduced? [duplicate]

Say a $\Bbbk$-algebra is separable if $L\otimes _\Bbbk A$ is reduced for every field extension $L/\Bbbk$, and reduced if its underlying ring is reduced. Separable always implied reduced, and I found ...
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### Prove every finite lattice has a greatest element - without induction

I have to prove that every finite lattice (L, ≤) has a greatest element. I have seen a lot of proofs proving this by using induction, however, I have to prove it without induction since our ...
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### Problem in solving a question related to roots of an equation.

The question is : Show that the equation $x^n+x^{n-1}+\cdots+x-1=0$ has unique positive root for all $n \in \mathbb {N}$ and all these positive roots lying in between $0$ and $1$ for all $n \geq 2$...
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### Given a finite Group G, with A, B subgroups prove the order of AB [closed]

How do you prove: Given a finite group $G$, with $A,B$ subgroups then $$|AB|=\frac{|A||B|}{|A \cap B|}.$$
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### Finite dimensional separable algebra is étale

Say a $\Bbbk$-algebra is separable if $L\otimes _\Bbbk A$ is reduced for every extension $L/\Bbbk$. Say it's étale if there's an extension $L/\Bbbk$ such that $L\otimes_\Bbbk A\cong \prod_1^nL$. Here'...
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### A finite abelian group containing a non-trivial subgroup which lies in every non-trivial subgroup is cyclic

Let $G$ be a finite abelian group s.t. it contains a subgroup $H_{0} \neq (e)$ which lies in every subgroup $H \neq (e)$. Prove that $G$ must be cyclic. Also what can be said about $o(G)$ ? I'm ...
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### $A$ a subset of a finite group $G$ with strictly more than $|G|/2$ elements. Show $AA=G$. [closed]

The question asks (a) Let $A$ be a subset of finite group $G$ with strictly greater than $|G|/2$ elements. Show $AA=G$ and (b) Show this can fail in a monoid. I've been working on this for awhile ...
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### On direct sum and direct product of groups

I understood the definitions of direct sums and direct products of groups (rings, modules, etc). The definitions of them can be given in terms of universal properties - certain commutative diagram in ...
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### The elements of finite order in an abelian group form a subgroup: proof check

If G is an abelian group, show that the set of elements of finite order is a subgroup of G. Proof: Let G be an abelian group and H be the set of elements of finite order. (1) nonempty Now e ∈ H, ...
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### $F(a_1,\ldots,a_n) = F(a_1+t_2 a_n+\cdots+t_n a_n)$ if Extension is Simple?

Suppose that F is an infinite field and that $L = F(a_1,\ldots,a_n)$ is a simple extension of $F$. Can you show, without the assumption that $[L:F] < \infty$ that $\exists t_2,\ldots,t_n \in F$ ...
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