## i have a final exam

alsalam alekum

hi

i need a help in some question

i try but i can not solve it

i have a final exam tomorrow

i need help in QUESTION TWO (b)

and QUESTION SIX (a)

if i solve another questions , i will return back to ask you

Thanks
bye
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 this question Q(2)(a) show the polynomail is irreducible over Q xpower 5+8*(x power 4)+3*(x power 2)+4*x+7 question six(a): Let F subset of K be a field extension show that |K:F|=1 if and only if K=F.

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 Quote by uob_student this question Q(2)(a) show the polynomail is irreducible over Q xpower 5+8*(x power 4)+3*(x power 2)+4*x+7
By the "rational root" theorem, what are the possible rational roots of this polynomial (there are only 4)? Are any of those actually roots? And what does "irreducible" MEAN?

 question six(a): Let F subset of K be a field extension show that |K:F|=1 if and only if K=F.
What is the DEFINITION of |K:F|?

## i have a final exam

 Quote by HallsofIvy By the "rational root" theorem, what are the possible rational roots of this polynomial (there are only 4)? Are any of those actually roots? And what does "irreducible" MEAN? What is the DEFINITION of |K:F|?

irreducible means that the polynomail do not need factorization

|k:F| means the degree of k over F

i need the answer today if you can
 hi i have another question: QUESTION:Let F:R________>S be an epimorphism of rings with unity (a) Show that if R is a principle ideal domain ,then every ideal in S is principle (b) Show by an example that S need not be an integral domain. thanks
 Recognitions: Homework Help Science Advisor That isn't what irreducible means,: polynomials do not 'need' or 'want' to be factored. You were asked to define the degree of a field extension, not write out the name again. Hint: the dimension of *** over *** as a ***** ***** what have you done for the last post? Have you written out the definition of a PID and tried to see what you can show? (b) is just trivial - what is the first kind of non-integral domain you ever meet (Hint: the smallest one has 4 elements). I presume that in (b) there are some extra conditions like R ought to still be assumed a PID or even an integral domain. Otherwise the identity morphism on any non-integral domain would do, wouldn't it?
 Recognitions: Homework Help Actually, Irreducible means that the polynomials cannot be simplified to another of lower degree, example $$x^4 + x^2 + 6$$ can be simplified by letting $x^2=a$ then you can reduce it to $$a^2 + a +6$$, which can be solved easily with the quadratic formula. Then once you have your $a$ values, you set them equal to $a$ and solve for x. Edit: Matt beat me to it
 Recognitions: Homework Help Science Advisor No. That is very very incorrect, Gib. Irreducible (over a field F) means something entirely different - cannot be written as a product of factors over strictly lower degree with coefficients in F. The polynomial you wrote down is irreducible over Q (note all questions of reducibility have some implicit underlying field)

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