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?

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 $S_3 \times S_2$ ?

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 […]

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.

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.

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 […]

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? 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.

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

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} = […]

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