chezmarylou1i

2022-01-01

Prove ${x}^{6}-6{x}^{4}+12{x}^{2}-11$ is irreducible over $\mathbb{Q}$

turtletalk75

Beginner2022-01-02Added 29 answers

Step 1

Let

Since none of the three elements 0,1,2 in

or

From the first equation, since corresponding coefficients are equal, we have

1)

2)

3)

4)

5)

From (1),

Consequently,

We have just shown that

For the second equation, however,

Hattie Schaeffer

Beginner2022-01-03Added 37 answers

Let

Substitute

Letteing

karton

Expert2022-01-09Added 613 answers

Step 1

If you are comfortable with a little more abstract algebra, here is an approach that does not require such ad hoc calculations. First it is easy to see that in

where

A quick check shows that h has no roots in

If p has an irreducible cubic factor in

As we saw before, the degree of this factor is even, so it is either quartic or sextic. Again, if it is sextic then p is irreducible in

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