Articles of abstract algebra

Example of a ring with prime characteristic, which is not an integral domain

We know that every intgral domain has prime (or 0) characteristic. Is there en example that the converse isn’t true? Does there exist a ring, which is not an integral domain, but has a prime characteristic?

The Galois group of two irreducible polynomials

What information is necessary to determine the Galois group of two irreducible polynomials? If I know the Galois group of $p(x)$ is $S_3$ and the Galois group of $f(x)$ is $S_2$ can I say the Galois group of $p(x)f(x)$ is $S_3 \times S_2$ ?

Question about notation for Minors / How to determine a set of invariant factors given a relations matrix

I have been trying to work some problems about determining a set of invariant factors given a relations matrix (in the sense of Jacobson) and vice versa. I am stuck and not sure if I am using the right tools. I have a couple questions but first I will state the relevant definitions and facts […]

Irreducible polynomial over integers : sufficient conditions?

Why is $9x^{2}-3$ reducible over integers? I am not able to understand what are the necessary and sufficient condition for the irreducibility of a polynomial.

Cubic with repeated roots has a linear factor

If $f$ is a cubic polynomial with a repeated root over a field then $f$ has a linear factor. I think that is true in perfect fields but I don’t know how to prove it.

Norms in $\mathbb{Q}$

So I ran through this problem in my study, and need a bit of clarification. Theorem: Let $r+si \in \mathbb{Q}[i]$. Then there is an element $a+bi \in \mathbb{Z}[i]$ such that $N((r+si)-(a+bi)) <1$. Proof: Pick integer $a$ and $b$ such that $|r-a| \leq 1/2$ and $|s-b| \leq 1/2$. Now we have $N((r+si)-(a+bi)) = N((r-a)+(s-b)i)$. Then: $=(r-a)^2+(s-b)^2 […]

On the fields of rational fractions over $\mathbb{F}_{p}$

Let $K$ be a field with characteristic $p>0$. Let $K^{p}=\left\{ a^{p}:a\in K\right\}$. Prove that $K^p$ is a subfield of $K$. Furthermore, if $K=\mathbb{F}_{p}\left(X\right)$ is the fields of rational fractions over $\mathbb{F}_{p}$, find $K^p$ and $[K:K^p]$. Help me prove the second part. Thanks a lot.

Is any homomorphism between two isomorphic fields an isomorphism?

Is any homomorphism between two isomorphic fields an isomorphism? What I mean is that two fields are called isomorphic if there exist one homomorphism between them . But not sure if existence of one isomorphism means there exist no non-bijective homomorphism between . An explanation or a counter example would help . Please.

Determine order of a Sylow p-subgroup

Let $G$ be a group of order $260$. For each prime $p$ dividing $|G|$, determine the order of a Sylow $p$-subgroup. We have $|G| = 260 = 2^2 \cdot 5 \cdot 13$.

$X_{2n}$ be group presentation as displayed below proof verification

Hi so I am solving problems in dummit and foote, however this problem I am not able to do it Show that if $n = 3k$, then $X_{2n}$ has order 6, and it has same generators and relations as $D_6$ when x is replaced by r and y is replaced by s. where $X_{2n} = […]