hi, everybody ! T = x^5 +3x^2 +2 is reductible in Q ?


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Question Number 147364 by henderson last updated on 20/Jul/21

$hi, everybody!$ $T = x^{5} + 3 x^{2} + 2 is reductible in Q ?$
	Answered by Olaf_Thorendsen last updated on 20/Jul/21

	$T \in Q [X], T (x) = x^{5} + 3 x^{2} + 2$ $T (x) = x^{5} + 3 x^{2} + 2$ ${It}^{'} s easy to verify with the sign of$ $T^{'} (x) that T has only one real root .$ $Then T is not reductible in Q, in R .$ $Only in C .$
	Answered by puissant last updated on 20/Jul/21

	$P i s a p o l y n o m i a l . .$ $P (X) = a_{n} X^{n} + a_{n - 1} X^{n - 1} + . . . . + a_{1} X + a_{0}$ $\exists p (p r i m e n u m b e r) /$ $∙ p s p l i t a_{0}, a_{1}, . . . . ., a_{n - 1}$ $∙ p n o s p l i t a_{n}$ $∙ p^{2} n o s p l i t a_{0}$ $\Rightarrow P i s i r r e d u c t i b l e i n Q [x]$ $(E i n s e n t e i n t h e o r e m) . . .$
	Commented by Olaf_Thorendsen last updated on 20/Jul/21

	$T h e E i s e n s t e i n i r r e d u c t i b i l i t y$ $c r i t e r i o n i s n o t v e r i f i e d h e r e m i s t e r .$
	Commented by puissant last updated on 20/Jul/21

	$m o n s i e u r c a v e u t d i r e q u e l e p o l y n o m e$ $e s t r e d u c t i b l e . .$
	Commented by Olaf_Thorendsen last updated on 20/Jul/21

	$N o n . S i l e c r i t e r e e s t v e r i f i e a l o r s l e$ $p o l y n o m e e s t i r r e d u c t i b l e d a n s Q e t$ $d o n c d a n s Z s e l o n l e l e m m e d e G a u s s .$ $S i l e c r i t e r e n^{'} e s t p a s r e s p e c t e, o n n e$ $p e u t r i e n d e d u i r e .$ $P a r e x e m p l e 4 x^{2} - 1 e s t r e d u c t i b l e$ $d a n s Q m a i s n e r e s p e c t e p a s l e$ $c r i t e r e .$
	Commented by puissant last updated on 20/Jul/21

	$O k e y m o n s i e u r j e c o m p r e n d m a i n t e n a n t$ $m e r c i p r o f . .$
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