4 Element by 4 Element Multiplication Table ....

[Math Amateur]

What would be the easiest (but neat and readable) way of entering a 4 element by 4 element multiplication table ... ... for example the multiplication table for the field of 4 elements ... ...

Peter

 

[MarkFL]

Do you mean something like this?

\(\displaystyle \begin{array}{|c|c|c|c|c|} \hline & 1 & 2 & 3 & 4 \\ \hline 1 & 1 & 2 & 3 & 4 \\ \hline 2 & 2 & 4 & 6 & 8 \\ \hline 3 & 3 & 6 & 9 & 12 \\ \hline 4 & 4 & 8 & 12 & 16 \\ \hline \end{array}\)

 

[Math Amateur]

Do you mean something like this?

\(\displaystyle \begin{array}{|c|c|c|c|c|} \hline & 1 & 2 & 3 & 4 \\ \hline 1 & 1 & 2 & 3 & 4 \\ \hline 2 & 2 & 4 & 6 & 8 \\ \hline 3 & 3 & 6 & 9 & 12 \\ \hline 4 & 4 & 8 & 12 & 16 \\ \hline \end{array}\)

Yes ... thanks Mark ...

BUT ... would be so much better if there was a darker line (or some distinguishing line around first row and first colum to distinguish elements of the finite field from the results of multiplying them ...

Maybe I am being obsessively neat ... ...:)

Peter

 

[MarkFL]

Yes ... thanks Mark ...

BUT ... would be so much better if there was a darker line (or some distinguishing line around first row and first colum to distinguish elements of the finite field from the results of multiplying them ...

Maybe I am being obsessively neat ... ...:)

Peter

I do agree that would be better, but I honestly don't know how to get MathJax to do that. :D

 

[Opalg]

BUT ... would be so much better if there was a darker line (or some distinguishing line around first row and first colum to distinguish elements of the finite field from the results of multiplying them ...

One way is to keep only those two lines and remove all the others: $$\begin{array}{c|cccc} + & a&b&c&d \\ \hline a&a&b&c&d\\ b&b&a&d&c \\ c&c&d&a&b \\ d&d&c&b&a \end{array}.$$

 

[Math Amateur]

I do agree that would be better, but I honestly don't know how to get MathJax to do that. :D

Thanks anyway Mark ...

Not many occasions when that happens ...

Peter

- - - Updated - - -

One way is to keep only those two lines and remove all the others: $$\begin{array}{c|cccc} + & a&b&c&d \\ \hline a&a&b&c&d\\ b&b&a&d&c \\ c&c&d&a&b \\ d&d&c&b&a \end{array}.$$

Thanks Opalg ... yes, definitely one solution...

Peter

 

[I like Serena]

How about:
\begin{tikzpicture}
\usetikzlibrary{matrix}
\matrix (m) [nodes={minimum width=2em,minimum height=2ex},matrix of nodes]
{
+&0&1&2&3\\
0&0&1&2&3\\
1&1&2&3&10\\
2&2&3&10&11\\
3&3&10&11&12\\
};
\draw[very thick] (m-1-1.north east) -- (m-5-1.south east);
\draw[very thick] (m-1-1.south west) -- (m-1-5.south east);
\foreach \x in {2,...,5}{
\draw (m-1-\x.north east) -- (m-5-\x.south east);
\draw (m-\x-1.south west) -- (m-\x-5.south east);
}
\end{tikzpicture}

 

[Math Amateur]

How about:
\begin{tikzpicture}
\usetikzlibrary{matrix}
\matrix (m) [nodes={minimum width=2em,minimum height=2ex},matrix of nodes]
{
+&0&1&2&3\\
0&0&1&2&3\\
1&1&2&3&10\\
2&2&3&10&11\\
3&3&10&11&12\\
};
\draw[very thick] (m-1-1.north east) -- (m-5-1.south east);
\draw[very thick] (m-1-1.south west) -- (m-1-5.south east);
\foreach \x in {2,...,5}{
\draw (m-1-\x.north east) -- (m-5-\x.south east);
\draw (m-\x-1.south west) -- (m-\x-5.south east);
}
\end{tikzpicture}

Thanks I like Serena ... excellent ...

... but ... takes a bit of effort ... :)

Peter

 

[I like Serena]

Thanks I like Serena ... excellent ...

... but ... takes a bit of effort ... :)

Peter

The matrix itself is exactly the same as it is in regular $\LaTeX$.
And we can embellish it using TikZ, which is really a bit of copy-and-paste.
If you're going to create more than one or two tables, we can probably come up with an easy way to do it for all of them.

 

[Math Amateur]

The matrix itself is exactly the same as it is in regular $\LaTeX$.
And we can embellish it using TikZ, which is really a bit of copy-and-paste.
If you're going to create more than one or two tables, we can probably come up with an easy way to do it for all of them.

OK

I tried this in my post but the multiplication table failed to display ... can anyone see my error\(\displaystyle \begin{tikzpicture}
\usetikzlibrary{matrix}
\matrix (m) [nodes={minimum width=2em,minimum height=2ex},matrix of nodes]
{
\times & 0 & 1 & \theta & \theta + 1 \\
0&0&0&0&0\\
1&0&1& \theta & \theta + 1 \\
\theta & 0 & \theta & \theta + 1 & 1 \\
\theta +1 & 0 & \theta + 1 & 1 & \theta \\
};
\draw[very thick] (m-1-1.north east) -- (m-5-1.south east);
\draw[very thick] (m-1-1.south west) -- (m-1-5.south east);
\foreach \x in {2,...,5}{
\draw (m-1-\x.north east) -- (m-5-\x.south east);
\draw (m-\x-1.south west) -- (m-\x-5.south east);
}
\end{tikzpicture}\)

 

[MarkFL]

You need to wrap any $\LaTeX$ in dollar-sign delimiters, however then an error regarding the matrix m being unrecognized crops up, and I don't see why. :D

 

[Math Amateur]

You need to wrap any $\LaTeX$ in dollar-sign delimiters, however then an error regarding the matrix m being unrecognized crops up, and I don't see why. :D

Thanks Mark ...

It is a real puzzle ... I did put \(\displaystyle delimiters around the whole of the Latex Text in my most recent post entitled[h=1] http://mathhelpboards.com/linear-abstract-algebra-14/field-extensions-finite-fields-dummit-foote-section-13-2-exercise-2-a-21441.html in the Linear and Abstract Algebra forum ... [/h]
Hope someone can figure out what is wrong ...

All I did was replace I like Serena's numbers with the values of the finite field multiplication table ...

Remains a mystery what went wrong ...

... BUT ... thanks again for your help ...

Peter\)

 

[I like Serena]

OK

I tried this in my post but the multiplication table failed to display ... can anyone see my error\(\displaystyle \begin{tikzpicture}
\usetikzlibrary{matrix}
\matrix (m) [nodes={minimum width=2em,minimum height=2ex},matrix of nodes]
{
\times & 0 & 1 & \theta & \theta + 1 \\
0&0&0&0&0\\
1&0&1& \theta & \theta + 1 \\
\theta & 0 & \theta & \theta + 1 & 1 \\
\theta +1 & 0 & \theta + 1 & 1 & \theta \\
};
\draw[very thick] (m-1-1.north east) -- (m-5-1.south east);
\draw[very thick] (m-1-1.south west) -- (m-1-5.south east);
\foreach \x in {2,...,5}{
\draw (m-1-\x.north east) -- (m-5-\x.south east);
\draw (m-\x-1.south west) -- (m-\x-5.south east);
}
\end{tikzpicture}\)

When we put [MATH] tags or \$ symbols around $\LaTeX$, we put latex in so called math mode, so that it's easier to create mathematical formulas.

With [TIKZ] tags or \begin {tikzpicture} ... \end {tikzpicture} we put it in TikZ mode, so that it's easier to make pictures.
Unfortunately, that means it's not in math mode.
So it's complaining about the \times directive that it does not recognize, and more or less suggesting to use \$ symbols.
We should put \$ symbols around the symbols that should be rendered in math mode.

\begin{tikzpicture}
\usetikzlibrary{matrix}

\matrix (m) [nodes={minimum width=3em,minimum height=3ex},matrix of nodes]
{
$\times$ & 0 & 1 & $\theta$ & $\theta + 1$ \\
0&0&0&0&0\\
1&0&1& $\theta$ & $\theta + 1$ \\
$\theta$ & 0 & $\theta$ & $\theta + 1$ & 1 \\
$\theta + 1$  & 0 & $\theta + 1$ & 1 & $\theta$ \\
};

\draw[very thick] (m-1-1.north east) -- (m-5-1.south east);
\draw[very thick] (m-1-1.south west) -- (m-1-5.south east);
\foreach \x in {2,...,5}{
  \draw (m-1-\x.north east) -- (m-5-\x.south east);
  \draw (m-\x-1.south west) -- (m-\x-5.south east);
}
\end{tikzpicture}

[TIKZ]
\usetikzlibrary{matrix}
\matrix (m) [nodes={minimum width=3em,minimum height=3ex},matrix of nodes]
{
$\times$ & 0 & 1 & $\theta$ & $\theta + 1$ \\
0&0&0&0&0\\
1&0&1& $\theta$ & $\theta + 1$ \\
$\theta$ & 0 & $\theta$ & $\theta + 1$ & 1 \\
$\theta + 1$ & 0 & $\theta + 1$ & 1 & $\theta$ \\
};
\draw[very thick] (m-1-1.north east) -- (m-5-1.south east);
\draw[very thick] (m-1-1.south west) -- (m-1-5.south east);
\foreach \x in {2,...,5}{
\draw (m-1-\x.north east) -- (m-5-\x.south east);
\draw (m-\x-1.south west) -- (m-\x-5.south east);
}
[/TIKZ]

 

[I like Serena]

Thanks Mark ...

It is a real puzzle ... I did put \(\displaystyle delimiters around the whole of the Latex Text in my most recent post entitled[h=1] http://mathhelpboards.com/linear-abstract-algebra-14/field-extensions-finite-fields-dummit-foote-section-13-2-exercise-2-a-21441.html in the Linear and Abstract Algebra forum ... [/h]
Hope someone can figure out what is wrong ...

All I did was replace I like Serena's numbers with the values of the finite field multiplication table ...

Remains a mystery what went wrong ...

... BUT ... thanks again for your help ...

Peter\)

\(\displaystyle

I've edited your post to fix the table.
That is, I've removed the superfluous \$ around it.
Instead I've put \$ symbols around \times and \theta where they occur.
And I've increased the minimum width in the matrix to 3em, so that the columns align properly.\)

 

[Math Amateur]

I've edited your post to fix the table.
That is, I've removed the superfluous \$ around it.
Instead I've put \$ symbols around \times and \theta where they occur.
And I've increased the minimum width in the matrix to 3em, so that the columns align properly.

Thank you for all your help ...

Your posts have been exceedingly helpful!

Peter

 

[I like Serena]

My previous TikZ suggestion had the 'problem' that math symbols had to have \$ symbols around them.
Turns out there is a better way by using a [M]matrix of math nodes[/M] instead of a [M]matrix of nodes[/M].
Then all nodes are already in math mode.
Other than that the solution is the same.
\begin{tikzpicture}
\usetikzlibrary{matrix}

\matrix (m) [nodes={minimum width=3em,minimum height=3ex},matrix of math nodes]
{
\times & 0 & 1 & \theta & \theta + 1 \\
0 & 0 & 0 & 0 & 0\\
1 & 0 & 1 & \theta & \theta + 1\\
\theta & 0 & \theta & \theta + 1 & 1 \\
\theta + 1 & 0 & \theta + 1 & 1 & \theta \\
};

\draw[very thick] (m-1-1.north east) -- (m-5-1.south east);
\draw[very thick] (m-1-1.south west) -- (m-1-5.south east);
\foreach \x in {2,...,5}{
\draw (m-1-\x.north east) -- (m-5-\x.south east);
\draw (m-\x-1.south west) -- (m-\x-5.south east);
}
\end{tikzpicture}
[latexs]\begin{tikzpicture}
\usetikzlibrary{matrix}

\matrix (m) [nodes={minimum width=3em,minimum height=3ex},matrix of math nodes]
{
\times & 0 & 1 & \theta & \theta + 1 \\
0 & 0 & 0 & 0 & 0\\
1 & 0 & 1 & \theta & \theta + 1\\
\theta & 0 & \theta & \theta + 1 & 1 \\
\theta + 1 & 0 & \theta + 1 & 1 & \theta \\
};

\draw[very thick] (m-1-1.north east) -- (m-5-1.south east);
\draw[very thick] (m-1-1.south west) -- (m-1-5.south east);
\foreach \x in {2,...,5}{
\draw (m-1-\x.north east) -- (m-5-\x.south east);
\draw (m-\x-1.south west) -- (m-\x-5.south east);
}
\end{tikzpicture}[/latexs]