# 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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### The square roots of different primes are linearly independent over the field of rationals

I need to find a way of proving that the square roots of a finite set of different primes are linearly independent over the field of rationals. I've tried to solve the problem using ...
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### Every nonzero element in a finite ring is either a unit or a zero divisor

Let $R$ be a finite ring with unity. Prove that every nonzero element of $R$ is either a unit or a zero-divisor.
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### Irreducibility of $X^{p-1} + \cdots + X+1$

Can someone give me a hint how to the irreducibility of $X^{p-1} + \cdots + X+1$, where $p$ is a prime, in $\mathbb{Z}[X]$ ? Our professor gave us already one, namely to substitute $X$ with $X+1$, ...
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### Why isn't an infinite direct product of copies of $\Bbb Z$ a free module?

Why isn't an infinite direct product of copies of $\Bbb Z$ a free module? Actually I was asked to show that it's not projective, but as $\Bbb{Z}$ is a PID, so it suffices to show it's not free. I ...
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### Submodule of free module over a p.i.d. is free even when the module is not finitely generated?

I have heard that any submodule of a free module over a p.i.d. is free. I can prove this for finitely generated modules over a p.i.d. But the proof involves induction on the number of generators, so ...
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### Show that every group of prime order is cyclic

Show that every group of prime order is cyclic. I was given this problem for homework and I am not sure where to start. I know a solution using Lagrange's theorem, but we have not proven Lagrange's ...
### Why can we use the division algorithm for $x-a$?
In Theorem 5.2.3 in these notes, it is said that Since $x − a$ has leading coefficient $1$, which is a unit, we may use the Division Algorithm... Why is this true? I thought that the Division ...
### The number of ring homomorphisms from $\mathbb{Z}_m$ to $\mathbb{Z}_n$
I face the problem of finding how many non-trivial ring or group homomorphisms there are from $\mathbb{Z}_m$ to $\mathbb{Z}_n$, where $m<n$. Is there any general formula? At the moment, I want to ...