show that if $K$ is a field then $K[x]$ is principal [duplicate]

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This question already has an answer here:




  • $F$ is a field iff $F[x]$ is a Principal Ideal Domain

    1 answer




I want to show that if $K$ is a field then $K[x]$ is principal



Here is what I did:



I took 2 polynomial p(x) and q(x). Since we're on a field we can do a euclidean division of polynomials:



$q(x) = p(x)a(x) + r(x)$ with $deg(r) < deg(p)$



Now if $r= 0$ we obviously have $p in (q)$



But I don't know how to show that it is principal if $r neq 0$



Also I would like to show that the assertion is false if we replace "$K$ is a field" by "$K$ is a principal domain and an integral domain". I can't figure out how to do that










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marked as duplicate by rschwieb abstract-algebra
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Mar 12 '17 at 16:03


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.














  • 1




    $begingroup$
    This and variants have been asked before. Check for example these: math.stackexchange.com/q/1214667/133781 math.stackexchange.com/q/2014146/133781 and math.stackexchange.com/q/873088/133781
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    – Xam
    Mar 12 '17 at 16:02
















1












$begingroup$



This question already has an answer here:




  • $F$ is a field iff $F[x]$ is a Principal Ideal Domain

    1 answer




I want to show that if $K$ is a field then $K[x]$ is principal



Here is what I did:



I took 2 polynomial p(x) and q(x). Since we're on a field we can do a euclidean division of polynomials:



$q(x) = p(x)a(x) + r(x)$ with $deg(r) < deg(p)$



Now if $r= 0$ we obviously have $p in (q)$



But I don't know how to show that it is principal if $r neq 0$



Also I would like to show that the assertion is false if we replace "$K$ is a field" by "$K$ is a principal domain and an integral domain". I can't figure out how to do that










share|cite|improve this question









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marked as duplicate by rschwieb abstract-algebra
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Mar 12 '17 at 16:03


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.














  • 1




    $begingroup$
    This and variants have been asked before. Check for example these: math.stackexchange.com/q/1214667/133781 math.stackexchange.com/q/2014146/133781 and math.stackexchange.com/q/873088/133781
    $endgroup$
    – Xam
    Mar 12 '17 at 16:02














1












1








1





$begingroup$



This question already has an answer here:




  • $F$ is a field iff $F[x]$ is a Principal Ideal Domain

    1 answer




I want to show that if $K$ is a field then $K[x]$ is principal



Here is what I did:



I took 2 polynomial p(x) and q(x). Since we're on a field we can do a euclidean division of polynomials:



$q(x) = p(x)a(x) + r(x)$ with $deg(r) < deg(p)$



Now if $r= 0$ we obviously have $p in (q)$



But I don't know how to show that it is principal if $r neq 0$



Also I would like to show that the assertion is false if we replace "$K$ is a field" by "$K$ is a principal domain and an integral domain". I can't figure out how to do that










share|cite|improve this question









$endgroup$





This question already has an answer here:




  • $F$ is a field iff $F[x]$ is a Principal Ideal Domain

    1 answer




I want to show that if $K$ is a field then $K[x]$ is principal



Here is what I did:



I took 2 polynomial p(x) and q(x). Since we're on a field we can do a euclidean division of polynomials:



$q(x) = p(x)a(x) + r(x)$ with $deg(r) < deg(p)$



Now if $r= 0$ we obviously have $p in (q)$



But I don't know how to show that it is principal if $r neq 0$



Also I would like to show that the assertion is false if we replace "$K$ is a field" by "$K$ is a principal domain and an integral domain". I can't figure out how to do that





This question already has an answer here:




  • $F$ is a field iff $F[x]$ is a Principal Ideal Domain

    1 answer








abstract-algebra ring-theory field-theory






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asked Mar 12 '17 at 15:56









Sylvester StalloneSylvester Stallone

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515311




marked as duplicate by rschwieb abstract-algebra
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Mar 12 '17 at 16:03


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.









marked as duplicate by rschwieb abstract-algebra
Users with the  abstract-algebra badge can single-handedly close abstract-algebra questions as duplicates and reopen them as needed.

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Mar 12 '17 at 16:03


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.










  • 1




    $begingroup$
    This and variants have been asked before. Check for example these: math.stackexchange.com/q/1214667/133781 math.stackexchange.com/q/2014146/133781 and math.stackexchange.com/q/873088/133781
    $endgroup$
    – Xam
    Mar 12 '17 at 16:02














  • 1




    $begingroup$
    This and variants have been asked before. Check for example these: math.stackexchange.com/q/1214667/133781 math.stackexchange.com/q/2014146/133781 and math.stackexchange.com/q/873088/133781
    $endgroup$
    – Xam
    Mar 12 '17 at 16:02








1




1




$begingroup$
This and variants have been asked before. Check for example these: math.stackexchange.com/q/1214667/133781 math.stackexchange.com/q/2014146/133781 and math.stackexchange.com/q/873088/133781
$endgroup$
– Xam
Mar 12 '17 at 16:02




$begingroup$
This and variants have been asked before. Check for example these: math.stackexchange.com/q/1214667/133781 math.stackexchange.com/q/2014146/133781 and math.stackexchange.com/q/873088/133781
$endgroup$
– Xam
Mar 12 '17 at 16:02










1 Answer
1






active

oldest

votes


















1












$begingroup$

Hint: Let $0 neq I subsetneq K[x]$ be an ideal. Let $0neq fin I$ such that



$$ deg(f)= min{ deg(g) : gin I }.$$



Use your argument above to show $I=(f)$.



For your second question I'd suggest that you show that the ideal $(2, x)subseteq mathbb{Z}[x]$ is not principal.






share|cite|improve this answer









$endgroup$




















    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    1












    $begingroup$

    Hint: Let $0 neq I subsetneq K[x]$ be an ideal. Let $0neq fin I$ such that



    $$ deg(f)= min{ deg(g) : gin I }.$$



    Use your argument above to show $I=(f)$.



    For your second question I'd suggest that you show that the ideal $(2, x)subseteq mathbb{Z}[x]$ is not principal.






    share|cite|improve this answer









    $endgroup$


















      1












      $begingroup$

      Hint: Let $0 neq I subsetneq K[x]$ be an ideal. Let $0neq fin I$ such that



      $$ deg(f)= min{ deg(g) : gin I }.$$



      Use your argument above to show $I=(f)$.



      For your second question I'd suggest that you show that the ideal $(2, x)subseteq mathbb{Z}[x]$ is not principal.






      share|cite|improve this answer









      $endgroup$
















        1












        1








        1





        $begingroup$

        Hint: Let $0 neq I subsetneq K[x]$ be an ideal. Let $0neq fin I$ such that



        $$ deg(f)= min{ deg(g) : gin I }.$$



        Use your argument above to show $I=(f)$.



        For your second question I'd suggest that you show that the ideal $(2, x)subseteq mathbb{Z}[x]$ is not principal.






        share|cite|improve this answer









        $endgroup$



        Hint: Let $0 neq I subsetneq K[x]$ be an ideal. Let $0neq fin I$ such that



        $$ deg(f)= min{ deg(g) : gin I }.$$



        Use your argument above to show $I=(f)$.



        For your second question I'd suggest that you show that the ideal $(2, x)subseteq mathbb{Z}[x]$ is not principal.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Mar 12 '17 at 16:02









        Severin SchravenSeverin Schraven

        6,1481934




        6,1481934















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            UmyeZJ2dO X,i4zB5iRtaoKZdQF DHfRocC FadjGG9g pc QR TbYshZPNZYCCSAwl,Ob1UW,4,2,OJRZ7Aub92j,7kxjRsWGO

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