Suppose $ L $ be an extension over $ \mathbb{Q} $ of degree $ n $. Let $\{e_{1},e_{2},\dots,e_{n}\} $ be a basis of this extension. Now I know the product $ e_{i}^{2} $ and $ e_{i}e_{j} $ . So we can determine the product $ a *b $ where $ a,b \in L $. But the calculation will so laborious. Is there any computational tool in sage math/programming/Matlab or something else to calculate this product? You can fix $ n $ like $ 6 $ or $8 $ .
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4$\begingroup$ Sage supports finite-dimensional algebras defined by multiplication tables for the basis elements. Is this what you want? $\endgroup$– Max AlekseyevCommented Jan 27, 2022 at 18:08
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$\begingroup$ Yes here is finite dimensional algebra over $\mathbb Q$ and multiplication table are given. So can we determine the product of arbitrary two elements? $\endgroup$– SkyCommented Jan 27, 2022 at 18:15
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2$\begingroup$ Yes. See example in the documentation. $\endgroup$– Max AlekseyevCommented Jan 27, 2022 at 18:21
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2$\begingroup$ I changed title and tags to match content. Please polish the main text, e.g. the second sentence is incomplete. $\endgroup$– GH from MOCommented Jan 27, 2022 at 20:14
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$\begingroup$ As I understand the question, the OP is considering a finite field extension $L$ of ${\mathbb Q}$ of degree $n$. Thus, there is a primitive element $\alpha\in L$ such that $1,\alpha,\ldots,\alpha^{n-1}$ is a basis of $L$ over ${\mathbb Q}$ and from that basis somehow the OP wants to compute or find formulas for the products $ab$, for general $a,b\in L$. For the degree $n=8$ that the OP specifically mentions at the end of the post, there are useful comments in the related post mathoverflow.net/questions/414814/… by the same OP. $\endgroup$– F ZaldivarCommented Jan 28, 2022 at 17:43
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