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Mathematics - Shifted Eisenstein

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 James Malloch 13 Jan 2014
To any mathematics boffins out there.

For Eisensteins Irreducibility Criterion you can perform a shift if the initial polynomial doesn't fit the criterion, preserving irreducibility.

Are there any trick for guessing what shift to perform? I normally do some long decision as it's just uni work so it is generally a shift of 1, -1 or 2 at most. So it's easy to check.

But generally, is there anything to help you?
Hi James,

I've not heard of the criterion before. But wiki

http://en.wikipedia.org/wiki/Eisenstein%27s_criterion#Advanced_explanation

seems to suggest that you can tell what prime you're after by looking at the discriminant of the polynomial. Then you can look at the polynomial mod p and try to tell what the shift should be.

It gives a nice example of a quadratic polynomial in which the shift is predictable.

Too tired to get the cubic case working...
OP James Malloch 14 Jan 2014
In reply to James Malloch:

Many thanks! I'd somehow missed that section when looking for it. I'll have a play about with it later.

Unfortunately, as most of the polynomials we seem to shift are degree 3 it will probably be as much work to get the determinant in an exam so just messing with a few shifts could be easier.

I will definitely look into it though! Thanks again!
In reply to James Malloch:

No problem. We rock-climbing Cumbrian mathematicians have to stick together!
OP James Malloch 14 Jan 2014
In reply to crossdressingrodney:

When you look at it like that, we're probably a rare breed!
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