Questions tagged [abstract-algebra]

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Why $\gcd(b,qb+r)=\gcd(b,r),\,$ so $\,\gcd(b,a) = \gcd(b,a\bmod b)$

Given: $a = qb + r$. Then it holds that $\gcd(a,b)=\gcd(b,r)$. That doesn't sound logical to me. Why is this so? Addendum by LePressentiment on 11/29/2013: (in the interest of http://meta.math....
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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 elementary ...
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Irreducible polynomial which is reducible modulo every prime

How to show that $x^4+1$ is irreducible in $\mathbb Z[x]$ but it is reducible modulo every prime $p$? For example I know that $x^4+1=(x+1)^4\bmod 2$. Also $\bmod 3$ we have that $0,1,2$ are not ...
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Prove that if $g^2=e$ for all $g$ in $G$ then $G$ is Abelian.

Prove that if $g^2=e$ for all $g$ in $G$ then $G$ is Abelian. This question is from group theory in Abstract Algebra and no matter how many times my lecturer teaches it for some reason I can't seem ...
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Why polynomial division algorithm works for $x-a$ or any monic polynomial?

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 ...
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Computing the Galois group of polynomials $x^n-a \in \mathbb{Q}[x]$

I have some problems with this exercise. I don't know if it can be done. Consider the polynomial $x^n - a \in \mathbb{Q}[x]$. Can I compute the Galois group of this over $\mathbb{Q}$? Maybe having ...
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Is an automorphism of the field of real numbers the identity map?

Is an automorphism of the field of real numbers $\mathbb{R}$ the identity map? If yes, how can we prove it? Remark An automorphism of $\mathbb{R}$ may not be continuous.
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Is $\mathbb{Q}(\sqrt{2}) \cong \mathbb{Q}(\sqrt{3})$?

In this post we saw isomorphism of vector spaces over $\mathbb{Q}$. Just came across this question: Is $\mathbb{Q}(\sqrt{2}) \cong \mathbb{Q}(\sqrt{3})$? I know these as $\mathbb{Q}$-Vector spaces, ...
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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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Denote by $M_{n \times n}(k)$ the ring of $n$ by $n$ matrices with coefficients in the field $k$. Then why does this ring not contain any two-sided ideal? Thanks for any clarification, and this is ...