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Join date: Jun 2, 2022

When a blank page turns into something more creative, and faster. CorelDRAW Graphics Suite 2020 software is a professional and complete suite of graphics design, editing, and publishing applications for Microsoft Windows, Mac. Total Pageviews, Add This Site to Facebook, Share to Twitter, Tweet and More! Downloads: 1232 - Price: Free Version: 7.0 Size: 2 MB Platform:.Q: How to determine the group order of a field extension $\mathbb{F}_p(X) / \mathbb{F}_p(X^q)$ Consider an extension $\mathbb{F}_p(X) / \mathbb{F}_p(X^q)$. How do you determine if it's a cyclic extension, what is the order of its Galois group and how do I compute this for a concrete example? A: $$\alpha x^q+bx+a=0\rightarrow \alpha x^q=bx=0$$ $$\Rightarrow x=\frac{1}{\alpha}=\frac{b}{a}$$ In general every polynomial is reduced so $a,b eq0$ if $\alpha$ is not in $\Bbb F_p$ then it's a cyclic extension. If $a=0$ then $b\in \Bbb F_p$ thus all coefficients are in $\Bbb F_p$; if $b=0$ then you obtain a linear extension thus $a eq 0$ then $b\in \Bbb F_p$. In order to compute the order of the Galois group you can use $\forall a \in \Bbb F_p^*, a^n=1, a^{p-1} eq 1\Rightarrow n=\frac{p-1}{d}$ where $d=deg(a)$; $$\mathbb{F}_p(X^q)/\mathbb{F}_p(X)=\left\{\frac{f(X)}{g(X)}\ |\ f,g\in \mathbb{F}_p(X)[X]\right\}$$ In the example $f(X)=X^q-1\in \mathbb{F}_p(X)$ and \$

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