How to show mutually orthogonal latin squares

Question

I have a question concerning mutually orthogonal latin squares (MOLS).

Let $ \mathbb F $ be a field of $n\in\mathbb N$ elements.

For all $q\in\mathbb F \backslash \{0\}$, define $n\times n $ tables $\mathit Q_q$ by $\mathit Q_q(x,y)=qx+y$.

Now I have to show, that those tables are MOLS. It seems pretty obvious, that those tables are MOLS. My Problem though is, that I am really new to combinatorics and hence failed to proof it...

I would appreciate any suggestion or little help. Thanks in advance.

Answer 1

Since there are exactly $n$ elements in your field, and your tables are $n\times n$, it suffices to show the following:

1. Every row of a single table has pairwise distinct elements.

2. Every column of a single table has pairwise distinct elements.

3. Every (row, column) pair of two tables has distinct elements.

To show (1), fix $q,y$. Suppose $Q_q(x,y)=Q_q(x',y)$, and prove that $x=x'$.

To show (2), fix $q,x$. Suppose $Q_q(x,y)=Q_q(x,y')$, and prove that $y=y'$.

To show (3), fix $x,y$. Suppose $Q_q(x,y)=Q_{q'}(x,y)$, and prove that $q=q'$.

Note: these are not combinatorial problems, they are problems in the algebra of finite fields.