The Rational Zero Theorem helps us to narrow down the list of possible rational zeros for a polynomial function. Once we have done this, we can use synthetic division repeatedly to determine all of the zeros of a polynomial function.
Suppose first that is a normal subgroup. Then by definition we must have for all , for every . Let and choose (). By hypothesis we have , i.e. for some , thus . So we have . You can prove in the same way.
Suppose for all . Let , we have to prove for every . So, let . We have that for some (by the hypothesis). hence we have . Because was chosen arbitrarily we have the desired
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