Timeline for Defining a sign of square roots in GF(p)
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Nov 26, 2023 at 0:46 | comment | added | Oleksandr Kulkov | I'd rather say $\mathbb Z$-module homomorphism. Non-trivial means that it does something meaningful, e.g. by mapping into corresponding square roots in $\mathbb F_{p^2}$, not just maps everything into 0. | |
| Nov 26, 2023 at 0:43 | comment | added | Daniel Asimov | "a non-trivial homomorphism" means a ring homomorphism, right? | |
| Nov 25, 2023 at 16:51 | comment | added | Oleksandr Kulkov | Ok, thanks! I meant the span in the answer. Amended it to reflect this. | |
| Nov 25, 2023 at 16:50 | history | edited | Oleksandr Kulkov | CC BY-SA 4.0 |
Z-span instead of ring extension
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| Nov 25, 2023 at 16:45 | comment | added | KConrad | As an example, $\mathbf Z[\sqrt{2},\sqrt{3}] = \mathbf Z + \mathbf Z\sqrt{2} + \mathbf Z\sqrt{3} + \mathbf Z\sqrt{6}$ while $\mathbf Z\sqrt{2} + \mathbf Z\sqrt{3}$ is smaller. | |
| Nov 25, 2023 at 16:43 | comment | added | KConrad | The brackets $[,]$ denote "ring". So $\mathbf Z[x,y,z]$ is the ring of polynomials in $x,y,z$ with integer coefficients, and the same notation applies with numbers in place of $x,y,z$, e.g., $\mathbf Z[\sqrt{6},\sqrt{10},\sqrt{15}]$ is all polynomial expressions in $\sqrt{6},\sqrt{10},\sqrt{15}$ with integer coefficients. Because they square to integers, we don't need powers of any square root above the 1st power. The $\mathbf Z$-span of $x,y,z$ is ${\mathbf Z}x + {\mathbf Z}y + {\mathbf Z}z$, and likewise with ${\mathbf Z}\sqrt{6} + {\mathbf Z}\sqrt{10} + {\mathbf Z}\sqrt{15}$. | |
| Nov 25, 2023 at 10:43 | comment | added | Oleksandr Kulkov | Sorry, I'm not very proficient with the notation. What I meant is literally the module composed of vectors $a \sqrt{6} + b\sqrt{10} + c\sqrt{15}$. I suppose $\mathbb Z[\sqrt{6}, \sqrt{10}, \sqrt{15}]$ is a different object which also has multiplication, but I want to only bother with addition here. Is there some concise notation and name for it? | |
| Nov 25, 2023 at 3:43 | comment | added | KConrad | Be careful: consider $\mathbf Z[\sqrt{6},\sqrt{10},\sqrt{15}]$. The numbers 6, 10, and 15 are all squarefree and distinct, but this ring does not have rank $3$ as $\mathbf Z$-module: it is a subring of finite index in the integers of the field $\mathbf Q(\sqrt{6},\sqrt{10},\sqrt{15})$, which is $\mathbf Q(\sqrt{6},\sqrt{10})$ since $6 \cdot 10 = 4 \cdot 15$. This is a field of degree $4$, not $8$, over $\mathbf Q$, and this implies $\mathbf Z[\sqrt{6},\sqrt{10},\sqrt{15}]$ has rank $4$ over $\mathbf Z$ (a $\mathbf Z$-basis is $\{1,\sqrt{6},\sqrt{10},\sqrt{15}\}$). | |
| Nov 25, 2023 at 3:25 | history | edited | KConrad | CC BY-SA 4.0 |
added 22 characters in body
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| Nov 25, 2023 at 1:17 | history | answered | Oleksandr Kulkov | CC BY-SA 4.0 |