MaBloWriMo 22: the order of omega, part II

Yesterday, from the assumption that s_{n-2} is divisible by M_n, we deduced the equations

\omega^{2^{n-1}} = q-1

and

\omega^{2^n} = 1

which hold in the group X^*. So what do these tell us about the order of \omega? Well, first of all, the second equation tells us that the order of \omega must be a divisor of 2^n, and the only divisors of 2^n are other powers of 2. So the order of \omega must be 2^k for some k \leq n.

Now suppose the order of \omega is 2^k, so \omega^{2^k} = 1. But then if we square both sides we get \omega^{2^{k+1}} = 1. Squaring again gives \omega^{2^{k+2}} = 1, and so on. So once we hit 1, we are stuck there: raising \omega to all bigger powers of 2 will also yield 1.

But now look at the first equation: \omega^{2^{n-1}} = q-1. Remember that the order of \omega has to be a power of 2. From this equation we can see that the order can’t be 2^{n-1}. Could it be a smaller power of two? In fact, no, it can’t, by the argument in the previous paragraph: once you hit a power of two that yields 1, all the higher powers also have to yield 1. So if \omega raised to any smaller power of 2 were the identity, then \omega^{2^{n-1}} would also have to be the identity—but it isn’t.

The inescapable conclusion is that the only possibility for the order of \omega is exactly 2^n.

So, how does that help? Hint: think about the order of the group X^*… the triumphant conclusion tomorrow!

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About Brent

Assistant Professor of Computer Science at Hendrix College. Functional programmer, mathematician, teacher, pianist, follower of Jesus.
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One Response to MaBloWriMo 22: the order of omega, part II

  1. Pingback: MaBloWriMo 23: contradiction! | The Math Less Traveled

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