text: # 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.
  
  8k views
  
  ### 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 ...
  3k views
  
  7k views
  
  6k views
  
  ### 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.
  1k views
  
  ### 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 $X+1$, ...
  1k views
  
  ### 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 ...
  4k views
  
  ### 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 ...
  5k views
  
  ### 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 ...
metadata: {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}