Can any splitting field always be regarded a splitting field of a irreducible polynomial

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Consider $mathbbQ(sqrt2,sqrt3)$ as the splitting field of $(X^2-2)(X^2-3)$ in $mathbbQ[X]$. On the other hand, $mathbbQ(sqrt2,sqrt3)=mathbbQ(sqrt2+sqrt3)$
and let



$X=sqrt2+sqrt3Rightarrow$



$X^2=5+2sqrt6Rightarrow$



$(X^2-5)^2=24Rightarrow$



$X^4-10X^2+1=0$



By Gauss' lemma, it is irreducible.



Is this true in general?




abstract-algebra galois-theory




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Consider $mathbbQ(sqrt2,sqrt3)$ as the splitting field of $(X^2-2)(X^2-3)$ in $mathbbQ[X]$. On the other hand, $mathbbQ(sqrt2,sqrt3)=mathbbQ(sqrt2+sqrt3)$
and let



$X=sqrt2+sqrt3Rightarrow$



$X^2=5+2sqrt6Rightarrow$



$(X^2-5)^2=24Rightarrow$



$X^4-10X^2+1=0$



By Gauss' lemma, it is irreducible.



Is this true in general?




abstract-algebra galois-theory




share|cite|improve this question



















  • Don't delete posts that got an answer.
    – quid♦
    Aug 1 at 21:54












up vote
0
down vote

favorite









up vote
0
down vote

favorite











Consider $mathbbQ(sqrt2,sqrt3)$ as the splitting field of $(X^2-2)(X^2-3)$ in $mathbbQ[X]$. On the other hand, $mathbbQ(sqrt2,sqrt3)=mathbbQ(sqrt2+sqrt3)$
and let



$X=sqrt2+sqrt3Rightarrow$



$X^2=5+2sqrt6Rightarrow$



$(X^2-5)^2=24Rightarrow$



$X^4-10X^2+1=0$



By Gauss' lemma, it is irreducible.



Is this true in general?




abstract-algebra galois-theory




share|cite|improve this question











Consider $mathbbQ(sqrt2,sqrt3)$ as the splitting field of $(X^2-2)(X^2-3)$ in $mathbbQ[X]$. On the other hand, $mathbbQ(sqrt2,sqrt3)=mathbbQ(sqrt2+sqrt3)$
and let



$X=sqrt2+sqrt3Rightarrow$



$X^2=5+2sqrt6Rightarrow$



$(X^2-5)^2=24Rightarrow$



$X^4-10X^2+1=0$



By Gauss' lemma, it is irreducible.



Is this true in general?





abstract-algebra galois-theory





share|cite|improve this question










share|cite|improve this question




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asked Aug 1 at 20:06









Xavier Yang

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  • Don't delete posts that got an answer.
    – quid♦
    Aug 1 at 21:54
















  • Don't delete posts that got an answer.
    – quid♦
    Aug 1 at 21:54















Don't delete posts that got an answer.
– quid♦
Aug 1 at 21:54




Don't delete posts that got an answer.
– quid♦
Aug 1 at 21:54










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Let's say it is true under very general conditions. This is known as the primitive element theorem. It states that whenever we have a separable field extension that is of finite degree, it is generated by a single element (and hence the splitting field of the irreducible polynomial of that element).



See https://en.wikipedia.org/wiki/Primitive_element_theorem






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    Let's say it is true under very general conditions. This is known as the primitive element theorem. It states that whenever we have a separable field extension that is of finite degree, it is generated by a single element (and hence the splitting field of the irreducible polynomial of that element).



    See https://en.wikipedia.org/wiki/Primitive_element_theorem






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      Let's say it is true under very general conditions. This is known as the primitive element theorem. It states that whenever we have a separable field extension that is of finite degree, it is generated by a single element (and hence the splitting field of the irreducible polynomial of that element).



      See https://en.wikipedia.org/wiki/Primitive_element_theorem






      share|cite|improve this answer























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        up vote
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        Let's say it is true under very general conditions. This is known as the primitive element theorem. It states that whenever we have a separable field extension that is of finite degree, it is generated by a single element (and hence the splitting field of the irreducible polynomial of that element).



        See https://en.wikipedia.org/wiki/Primitive_element_theorem






        share|cite|improve this answer













        Let's say it is true under very general conditions. This is known as the primitive element theorem. It states that whenever we have a separable field extension that is of finite degree, it is generated by a single element (and hence the splitting field of the irreducible polynomial of that element).



        See https://en.wikipedia.org/wiki/Primitive_element_theorem







        share|cite|improve this answer













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        share|cite|improve this answer











        answered Aug 1 at 20:12









        AlgebraicsAnonymous

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