Tagged Questions

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Show two multiplication tables of GF(8) are isomorphic [duplicate]

How to show the two tables above are isomorphic? I try to map one element to another element in another table, but I fail to do so as I found that one element from the table on the left is mapped to ...
38 views

Minimal polynomial reducible modulo every prime $p$

Suppose $K = \mathbb{Q}(\alpha)$ with $\alpha = a + b\sqrt{D_1}+c\sqrt{D_2}+d\sqrt{D_1D_2}$ with $D_1,D_2 \in \mathbb{Z}$. Prove that the minimal polynomial $m_\alpha(x)$ for $\alpha$ over ...
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Does the element exist in the Galois Field?

Let p be a prime positive integer and let a be an element of GF(p). Does there necessarily exist an element b of GF(p) satisfying b^2=a? So, taking a element of GF(p), can we find a b element of ...
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$K\supset \mathbb{F}_q$, $h=x^q-x+a$, $a\in K$ if $K$ is finite $h$ is reducible

$q=p^n$, $K\supset \mathbb{F}_q$, $h=x^q-x+a\in K[x]$ if $K$ is finite $h$ is reducible. Let $L$ the splitting field of $h$ over $K$. Attempt: I proved that if $\beta$ is a root of $h$ and $h$ ...
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Two questions on finite fields

I'm having some difficult with finite fields. If someone could point out a direction in which to look for these, or link to relevant material online, I would really appreciate it! I'm asked to factor ...
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Galois group of algebraic closure of a finite field

Is there any element of finite order of the Galois group of algebraic closure of a finite field, and if there is how can I construct it ? Thanks.
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Irreducible factors for $x^q-x-a$ in $\mathbb{F}_p$.

If $\mathbb{F}_q$ is a finite field with $q=p^m$ elements where $p$ is prime, if $a\in\mathbb{F}_p^\times$, ¿Which are the irreducible factors of $f_a=x^q-x-a$ ? Attempt: If $\alpha$ is a root for ...
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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 ...
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Ring structure on the Galois group of a finite field

Let $F$ be a finite field. There is an isomorphism of topological groups $(\mathrm{Gal}(\overline{F}/F),\circ) \cong (\widehat{\mathbb{Z}},+)$. It follows that the Galois group carries the structure ...
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Galois extension

http://www.math.uiuc.edu/~r-ash/Algebra/Chapter6.pdf In the pdf, corollary $6.4.2$, the extension $E/F_p$ is Galois. Why is it Galois? Is it because $F_p$ is a finite field and hence every ...
In the first chapter of Algebraic Number Theory (lecture notes collected by Cassels-Fröhlich), page 28 has the following paragraph: "We suppose now that $k$ is a finite field of characteristic $p$ ...