Help with Final Exam Questions: QUESTION TWO (b) & SIX (a)

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In summary, your polynomial is irreducible over Q, meaning it cannot be factored into factors with rational coefficients.
  • #1
uob_student
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alsalam alekum

hi

i need a help in some question

i try but i can not solve it

i have a final exam tomorrow:frown:

i need help in QUESTION TWO (b)

and QUESTION SIX (a)

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

Thanks
bye
 
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  • #2
i forget questions

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.
 
  • #3
uob_student said:
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|?
 
  • #4
HallsofIvy said:
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 :frown:
 
  • #5
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
 
  • #6
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?
 
  • #7
Actually, Irreducible means that the polynomials cannot be simplified to another of lower degree, example [tex]x^4 + x^2 + 6[/tex] can be simplified by letting [itex]x^2=a[/itex] then you can reduce it to [tex]a^2 + a +6[/tex], which can be solved easily with the quadratic formula. Then once you have your [itex]a[/itex] values, you set them equal to [itex]a[/itex] and solve for x.

Edit: Matt beat me to it
 
  • #8
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)
 
  • #9
uob_student said:
irreducible means that the polynomail do not need factorization
Well, not "need" factorization, "CANNOT be factored", in this case into factors with rational coefficients.

The "rational root theorem" says any rational roots of the equation "ax^n+ ...+ cx+ b= 0" (zeroes of the polynomial), with integer coefficients, must be of the form m/n where m is a factor of b and m is a factor of a. Now answer the rest of my question: What are the possible rational zeroes of that polynomial according to the rational root theorem. Are those possible zeroes actually zeroes? What does that tell you about its factorization?

|k:F| means the degree of k over F
No, that's just the how you would say it in words. What is the definition of "degree of K over F"?
 
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  • #10
your polynomial is irreducible if it is irreducible mod 2.

there it suffices to see if it has either an irreducible linear or quadratic factor.

halls tip let's you check for linear factors, and there is only one irreducible quadratic polynomial mod 2.

then your polynomial is irreducible over Z hence also over Q.
 

Related to Help with Final Exam Questions: QUESTION TWO (b) & SIX (a)

1. How should I approach answering question two (b)?

It is important to carefully read the question and understand what is being asked. Then, make a list of key terms and concepts that are relevant to the question. Next, brainstorm possible solutions or approaches and select the most appropriate one. Finally, clearly and concisely explain your chosen approach in your answer.

2. Can I use outside sources to help answer question two (b)?

Yes, you can use outside sources such as textbooks, research papers, and online resources to help support your answer. However, be sure to properly cite any information or ideas that are not your own.

3. What is the best way to prepare for question six (a)?

To prepare for question six (a), review all relevant course material and lecture notes. Pay attention to any key concepts or theories that may be relevant to the question. It may also be helpful to practice solving similar problems or scenarios to familiarize yourself with the process.

4. Are there any specific strategies or techniques that can help me answer question six (a)?

One helpful strategy for question six (a) is to break down the problem into smaller, more manageable parts. This can help you identify any key information or variables and determine the appropriate steps to solve the problem. It may also be useful to draw diagrams or use visual aids to better understand the problem.

5. Can I ask my professor for clarification or guidance on these questions?

Yes, it is always a good idea to reach out to your professor if you have any questions or need clarification on the exam questions. They can provide further guidance and help ensure that you are on the right track with your answers.

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