What is “the set of all polynomials in $pi$”?












0














From Ian Stewart's Galois Theory (2015, 4e, p. 20):



enter image description here



What does, for example, "the set of all polynomials in $pi$" mean?






polynomials terminology







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  • 4




    For example, $pi^2-2pi+1$ or $7pi^5-3pi^2+2pi$
    – David Peterson
    Nov 25 at 3:29


















0














From Ian Stewart's Galois Theory (2015, 4e, p. 20):



enter image description here



What does, for example, "the set of all polynomials in $pi$" mean?






polynomials terminology







share|cite|improve this question




















  • 4




    For example, $pi^2-2pi+1$ or $7pi^5-3pi^2+2pi$
    – David Peterson
    Nov 25 at 3:29
















0












0








0







From Ian Stewart's Galois Theory (2015, 4e, p. 20):



enter image description here



What does, for example, "the set of all polynomials in $pi$" mean?






polynomials terminology







share|cite|improve this question















From Ian Stewart's Galois Theory (2015, 4e, p. 20):



enter image description here



What does, for example, "the set of all polynomials in $pi$" mean?






polynomials terminology




polynomials terminology






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Nov 26 at 13:52









Shaun

8,701113680




8,701113680










asked Nov 25 at 3:26









dtcm840

33614




33614








  • 4




    For example, $pi^2-2pi+1$ or $7pi^5-3pi^2+2pi$
    – David Peterson
    Nov 25 at 3:29
















  • 4




    For example, $pi^2-2pi+1$ or $7pi^5-3pi^2+2pi$
    – David Peterson
    Nov 25 at 3:29










4




4




For example, $pi^2-2pi+1$ or $7pi^5-3pi^2+2pi$
– David Peterson
Nov 25 at 3:29






For example, $pi^2-2pi+1$ or $7pi^5-3pi^2+2pi$
– David Peterson
Nov 25 at 3:29












4 Answers
4






active

oldest

votes


















3














The terminology is hinted at in $(4)$.



A polynomial $p$ in $x$, with, say, integer coefficients, is defined by $$p(x)=sum_{i=0}^{n}a_ix^i,$$ where $a_iinBbb Z$ for all $i$, and for some $ninBbb Ncup{0}$.



So let $x=pi$ . . .






share|cite|improve this answer























  • Thank you for correcting my typo, @LordSharktheUnknown.
    – Shaun
    Nov 25 at 6:31



















3














The set of all polynomials in $pi$ with rational coefficients is the set of real numbers of the form $p(pi)$, where $p(x)$ is a polynomial with rational coefficients; that is, it is the set ${ p(pi) mid p(x) in mathbb{Q}[x] }$.






share|cite|improve this answer





























    2














    A "polynomial in $pi$ with integer coefficient" is a slightly sloppy shorthand for "a number that is the value of some polynomial with integer coefficients, evaluated at $x=pi$".



    In other words, the set of all such numbers is the range of the evaluation morphism $mathbb Z[X]tomathbb C$ that maps $X$ to $pi$.






    share|cite|improve this answer





























      0














      A polynomial in $pi$ is in $4)$ an expression of the form $q_npi^n+q_{n-1}pi^{n-1}+dots +q_1pi+q_0$, where each $q_iinmathbb Q$.



      Or, as in $3)$, you could replace the $q_iinmathbb Q$ with $a_iinmathbb Z$, and have polynomials in $pi$ with integer coefficients.



      In these two cases we get a subring, but not a subfield, of $mathbb C$.



      On the other hand, we get a subfield when we consider all rational expressions in $pi$; that is, quotients of polynomials in $pi$ over $mathbb Q$.






      share|cite|improve this answer





















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        4 Answers
        4






        active

        oldest

        votes








        4 Answers
        4






        active

        oldest

        votes









        active

        oldest

        votes






        active

        oldest

        votes









        3














        The terminology is hinted at in $(4)$.



        A polynomial $p$ in $x$, with, say, integer coefficients, is defined by $$p(x)=sum_{i=0}^{n}a_ix^i,$$ where $a_iinBbb Z$ for all $i$, and for some $ninBbb Ncup{0}$.



        So let $x=pi$ . . .






        share|cite|improve this answer























        • Thank you for correcting my typo, @LordSharktheUnknown.
          – Shaun
          Nov 25 at 6:31
















        3














        The terminology is hinted at in $(4)$.



        A polynomial $p$ in $x$, with, say, integer coefficients, is defined by $$p(x)=sum_{i=0}^{n}a_ix^i,$$ where $a_iinBbb Z$ for all $i$, and for some $ninBbb Ncup{0}$.



        So let $x=pi$ . . .






        share|cite|improve this answer























        • Thank you for correcting my typo, @LordSharktheUnknown.
          – Shaun
          Nov 25 at 6:31














        3












        3








        3






        The terminology is hinted at in $(4)$.



        A polynomial $p$ in $x$, with, say, integer coefficients, is defined by $$p(x)=sum_{i=0}^{n}a_ix^i,$$ where $a_iinBbb Z$ for all $i$, and for some $ninBbb Ncup{0}$.



        So let $x=pi$ . . .






        share|cite|improve this answer














        The terminology is hinted at in $(4)$.



        A polynomial $p$ in $x$, with, say, integer coefficients, is defined by $$p(x)=sum_{i=0}^{n}a_ix^i,$$ where $a_iinBbb Z$ for all $i$, and for some $ninBbb Ncup{0}$.



        So let $x=pi$ . . .







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Nov 25 at 6:53

























        answered Nov 25 at 3:33









        Shaun

        8,701113680




        8,701113680












        • Thank you for correcting my typo, @LordSharktheUnknown.
          – Shaun
          Nov 25 at 6:31


















        • Thank you for correcting my typo, @LordSharktheUnknown.
          – Shaun
          Nov 25 at 6:31
















        Thank you for correcting my typo, @LordSharktheUnknown.
        – Shaun
        Nov 25 at 6:31




        Thank you for correcting my typo, @LordSharktheUnknown.
        – Shaun
        Nov 25 at 6:31











        3














        The set of all polynomials in $pi$ with rational coefficients is the set of real numbers of the form $p(pi)$, where $p(x)$ is a polynomial with rational coefficients; that is, it is the set ${ p(pi) mid p(x) in mathbb{Q}[x] }$.






        share|cite|improve this answer


























          3














          The set of all polynomials in $pi$ with rational coefficients is the set of real numbers of the form $p(pi)$, where $p(x)$ is a polynomial with rational coefficients; that is, it is the set ${ p(pi) mid p(x) in mathbb{Q}[x] }$.






          share|cite|improve this answer
























            3












            3








            3






            The set of all polynomials in $pi$ with rational coefficients is the set of real numbers of the form $p(pi)$, where $p(x)$ is a polynomial with rational coefficients; that is, it is the set ${ p(pi) mid p(x) in mathbb{Q}[x] }$.






            share|cite|improve this answer












            The set of all polynomials in $pi$ with rational coefficients is the set of real numbers of the form $p(pi)$, where $p(x)$ is a polynomial with rational coefficients; that is, it is the set ${ p(pi) mid p(x) in mathbb{Q}[x] }$.







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered Nov 25 at 3:30









            Clive Newstead

            50.4k474133




            50.4k474133























                2














                A "polynomial in $pi$ with integer coefficient" is a slightly sloppy shorthand for "a number that is the value of some polynomial with integer coefficients, evaluated at $x=pi$".



                In other words, the set of all such numbers is the range of the evaluation morphism $mathbb Z[X]tomathbb C$ that maps $X$ to $pi$.






                share|cite|improve this answer


























                  2














                  A "polynomial in $pi$ with integer coefficient" is a slightly sloppy shorthand for "a number that is the value of some polynomial with integer coefficients, evaluated at $x=pi$".



                  In other words, the set of all such numbers is the range of the evaluation morphism $mathbb Z[X]tomathbb C$ that maps $X$ to $pi$.






                  share|cite|improve this answer
























                    2












                    2








                    2






                    A "polynomial in $pi$ with integer coefficient" is a slightly sloppy shorthand for "a number that is the value of some polynomial with integer coefficients, evaluated at $x=pi$".



                    In other words, the set of all such numbers is the range of the evaluation morphism $mathbb Z[X]tomathbb C$ that maps $X$ to $pi$.






                    share|cite|improve this answer












                    A "polynomial in $pi$ with integer coefficient" is a slightly sloppy shorthand for "a number that is the value of some polynomial with integer coefficients, evaluated at $x=pi$".



                    In other words, the set of all such numbers is the range of the evaluation morphism $mathbb Z[X]tomathbb C$ that maps $X$ to $pi$.







                    share|cite|improve this answer












                    share|cite|improve this answer



                    share|cite|improve this answer










                    answered Nov 25 at 3:30









                    Henning Makholm

                    238k16303537




                    238k16303537























                        0














                        A polynomial in $pi$ is in $4)$ an expression of the form $q_npi^n+q_{n-1}pi^{n-1}+dots +q_1pi+q_0$, where each $q_iinmathbb Q$.



                        Or, as in $3)$, you could replace the $q_iinmathbb Q$ with $a_iinmathbb Z$, and have polynomials in $pi$ with integer coefficients.



                        In these two cases we get a subring, but not a subfield, of $mathbb C$.



                        On the other hand, we get a subfield when we consider all rational expressions in $pi$; that is, quotients of polynomials in $pi$ over $mathbb Q$.






                        share|cite|improve this answer


























                          0














                          A polynomial in $pi$ is in $4)$ an expression of the form $q_npi^n+q_{n-1}pi^{n-1}+dots +q_1pi+q_0$, where each $q_iinmathbb Q$.



                          Or, as in $3)$, you could replace the $q_iinmathbb Q$ with $a_iinmathbb Z$, and have polynomials in $pi$ with integer coefficients.



                          In these two cases we get a subring, but not a subfield, of $mathbb C$.



                          On the other hand, we get a subfield when we consider all rational expressions in $pi$; that is, quotients of polynomials in $pi$ over $mathbb Q$.






                          share|cite|improve this answer
























                            0












                            0








                            0






                            A polynomial in $pi$ is in $4)$ an expression of the form $q_npi^n+q_{n-1}pi^{n-1}+dots +q_1pi+q_0$, where each $q_iinmathbb Q$.



                            Or, as in $3)$, you could replace the $q_iinmathbb Q$ with $a_iinmathbb Z$, and have polynomials in $pi$ with integer coefficients.



                            In these two cases we get a subring, but not a subfield, of $mathbb C$.



                            On the other hand, we get a subfield when we consider all rational expressions in $pi$; that is, quotients of polynomials in $pi$ over $mathbb Q$.






                            share|cite|improve this answer












                            A polynomial in $pi$ is in $4)$ an expression of the form $q_npi^n+q_{n-1}pi^{n-1}+dots +q_1pi+q_0$, where each $q_iinmathbb Q$.



                            Or, as in $3)$, you could replace the $q_iinmathbb Q$ with $a_iinmathbb Z$, and have polynomials in $pi$ with integer coefficients.



                            In these two cases we get a subring, but not a subfield, of $mathbb C$.



                            On the other hand, we get a subfield when we consider all rational expressions in $pi$; that is, quotients of polynomials in $pi$ over $mathbb Q$.







                            share|cite|improve this answer












                            share|cite|improve this answer



                            share|cite|improve this answer










                            answered Nov 25 at 4:00









                            Chris Custer

                            10.8k3724




                            10.8k3724






























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