Splitting Fields

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Abstract Algebra › Splitting Fields

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What definition does the following correlate to?

If is a prime, then the following polynomial is irreducible over the field of rational numbers.

$\phi(x)=x^{p-1}+...+x+1$

Eisenstein's Irreducibility Criterion

Gauss's Lemma

Primitive Field Theorem

Ideals Theorem

Principal Ideal Domain

The Eisenstein's Irreducibility Criterion is the theorem for which the given statement is a corollary to.

The Eisenstein's Irreducibility Criterion is as follows.

$f(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_0$

is a polynomial with coefficients that are integers. If there is a prime number that satisfy the following,

$\a_{n-1}\equiv a_{n-2}\equiv ...\equiv a+0\equiv 0 (\mod b)) \a_n ot\equiv 0 (\mod b), a_0 ot\equiv 0 (\mod b^2)$

Then over the field of rational numbers is said to be irreducible.