documentclass[12pt]amsart usepackageeucal,amssymb par newedtheoremtheorem����[section]newedtheoremnitheorem���ޤ���פǤʤ�����[section]renewedcommandthenitheorem newedtheoremrefpropPropositionrenewedcommandtherefprop newedtheoremcor��[section]newedtheoremlemma����[section]newedtheoremfact����[section]newedtheoremproposition[theorem]̿��newedtheoremaxAxiompar theoremstyledefinition newedtheoremdfn���[section] newedtheoremexmp��[section] newedtheoremexample��[section]newedtheoremdefinition���[section] newedtheoremq����[section] newedtheoremexq����[section] newedtheoremkeywd���ؤΥ������par theoremstyleremark newedtheoremrem����[section]newedtheoremclaim[]renewedcommandtheclaim par numberwithinequationsection par newedcommandtheoremref[1]Theorem ref#1 newedcommandsecref[1]Sref#1 newedcommandlemref[1]Lemma ref#1 par newedcommandZmbox ${mathbb {Z}}$ newedcommandQmbox ${mathbb {Q}}$ newedcommandRmbox ${mathbb {R}}$ newedcommandNmbox ${mathbb {N}}$ newedcommandImbox ${sqrt {-1}}$ par newedcommandLeg[2]mbox newedcommandFpmbox ${mathbb F}_p$ newedcommandkpekembox$k^times$ newedcommandbigzerolsmashhboxhuge 0 newedcommandbigzerousmashlower1.7exhboxhuge 0 par setlengthtopmargin-1.5cm setlengthtextheight26cm begindocument renewedcommandthepage title[����� I No.thistime ]����� I No.thistime ����� quad vskip-3pc maketitle par setcountersectionthistime par noindent par newedcommandmymondaiI �Ľ�Ʊ��

$$varphi:Q [X] to Q times Q times Q$$

�ǡ� $varphi(X)=(1,2,3)$ ����������ΤˤĤ��� �ͤ������� ���Τ褦�ʤ�Τ����ä��Ȥ��ơ� beginenumerate item $varphi(frac{1}{6})$ �Ϥ�����ˤʤ�٤����������� item $varphi(X^4)$ �Ϥ�����ˤʤ�٤����������� item $p(X)in Q [X]$ �ˤ������ơ� $varphi(p)$ ����ʤ����� endenumerate par par newedcommandmymondaiII begintheorem_type[q][q][section][definition][][] $Q$ ���� $Z /7Z$ �ؤδĽ�Ʊ����¸�ߤ������������endtheorem_type par newedcommandmymondaiIII begintheorem_type[q][q][section][definition][][] �Ľ�Ʊ��

$$psi:Z ni nmapsto ([n]_{3},[n]_{5},[n]_{7})in Z /3Z times Z /5Z times Z /7Z$$

��ͤ��롣 (��������$[n]_{m}$ �� ���� $n$ �� $Z /mZ$ �ˤ����� ���饹�򤢤�魯��) par beginenumerate item $Ker (psi)$ ����衣 item �Ĥν�Ʊ�������ˤ�� $psi$ ����Ƴ�����ñ�ͽ�Ʊ�������� �ɤ�ʤ�Τ���(���ξ���˸¤��������ס�) item $3$ �� $5cdot 7$ �Ȥǥ桼����åɤθ߽�ˡ��Ԥʤ��� $psi(x)=([1]_3,[0]_5,[0]_7)$ ��ߤ��� $xin Z$ ���ĵ��ʤ����� item $psi(x)=([1]_3,[2]_5,[3]_7)$ ��ߤ��� $xin Z$ ���ĵ��ʤ����� item $3$ �dz��� $1$ ���ޤꡢ $5$ �dz��� $2$ ���ޤꡢ $7$ �dz��� $3$ ;��褦���������������3�ĵ󤲤ʤ����� endenumerate parendtheorem_type par ����: beginitemize item �������ߤϤʤ�Ǥ�ĤǤ��롣â���̿���ǽ����Ĥ�Τ䡢 ¾�ͤ����Ǥˤʤ��Τ������ item �����ѻ汦��ˤ�˺�줺�˳����ֹ��̾����񤯤��ȡ� item �����������Ǥ⡢��ʬ������(��ͳ)���񤤤Ƥ��ʤ������ˤĤ��Ƥ� ���Ȥ�����Ǥ��äƤ�ۤȤ��ɾ�����ʤ��� enditemize par begintheorem_type[q][q][section][definition][][] mymondaiIendtheorem_type mymondaiII mymondaiIII par pagebreak setcounterq0 begintheorem_type[q][q][section][definition][][] mymondaiIendtheorem_type par noindent ����: par noindent(1): beginalign* 6varphi(frac16) &= varphi(frac16) +varphi(frac16) +varphi(frac16) +varphi(frac16) +varphi(frac16) +varphi(frac16)
&=varphi( frac16 +frac16 +frac16 +frac16 +frac16 +frac16 ) =varphi(1)=(1,1,1) endalign* �椨�ˡ�

$$varphi(frac{1}{6})= ( frac{1}{6}, frac{1}{6}, frac{1}{6})$$

par noindent(2):

$$varphi(X^4)=varphi(X)^4=(1^4,2^4,3^4)=(1,16,81).$$

par noindent(3) (1) ��Ʊ�ͤˤ��ơ�

$$varphi(a)=(a,a,a) qquad (forall a in Q )$$

���狼�롣

$$p(X)=sum_j p_j X^j qquad (p_jin Q )$$

�Ƚ񤯤� beginalign* varphi(p)&=varphi(sum_j p_j X^j)
&=sum_j varphi(p_j) varphi(X)^j
&=sum_j (p_j,p_j,p_j)cdot (1,2,3)^j
&=(sum_j p_j 1^j , sum_j p_j 2^j, sum_j p_j 3^j)
&=(p(1),p(2),p(3)) endalign* �Ĥޤ�

$$varphi(p)=(p(1),p(2),p(3)).$$

par pagebreak mymondaiII par noindent ����:¸�ߤ��ʤ��� par noindent ��ͳ: �⤷���Τ褦�ʤ�� $phi$ �����ä��Ȥ���ȡ�

$$[1]_7=phi(1) =phi(frac{1}{7})phi(7)=phi(frac{1}{7})[7]_7=phi(frac{1}{7})[0]_7=[0]_7$$

�Ȥʤä�̷�⤹�뤫�顣 par mymondaiIII par noindent(����): par noindent(1): beginalign* Ker (psi) &=psi^-1([0 ]_3,[0]_5,[0]_7)
&={nin Z ; [n]_3=[0]_3 text ���� [n]_5=[0]_5 text ���� [n]_7=[0]_7 }
&= { nin Z ; nin 3Z text ���� nin 5Z text ���� nin 7Z }
&=3cdot 5cdot 7Z = 105Z endalign* par noindent(2):

$$overline{psi}: Z /105Z to Z /3Z times Z /5Z times Z /7Z ,$$

$$psi([n]_{105})=([n]_3,[n]_5,[n]_7)$$

noindent(3):

$$12times 3+(-1)times 35=1$$

�����Ѥ��롣

$$psi(-35)=([1]_3,[0]_5,[0]_7)$$

��� $x=-35$ �ϵ����Τΰ�ĤǤ��롣 par noindent(4): 5 �� $3cdot 7=21$ �ȤǸ߽�ˡ��Ԥʤ���

$$(-4)*5+1*21=1$$

�����롣���Ʊ�ͤˤ��� beginalign* psi(-35)&=([1]_3,[0]_5,[0]_7)
psi(21)&=([0]_3,[1]_5,[0]_7)
psi(15)&=([0]_3,[0]_5,[1]_7)
endalign* �򤨤롣(�Ǹ�μ��� $psi(1)=([1]_3,[1]_5,[1]_7)$ �Ⱦ��󼰤��������롣) par beginalign* &([1]_3,[2]_5,[3]_7)
=& 1([1]_3,[0]_5,[0]_7) +2([0]_3,[1]_5,[0]_7) +3([0]_3,[0]_5,[1]_7)
=&psi(-35) +2 psi(21)+3psi(15)
=&psi(-35+42+45)=psi(52) endalign* par �椨�� $x=52$ �ϰ�Ĥ���Ǥ��롣 par noindent(5): par

$$52,52+105(=157), 52+105*100(=10552)$$

�ʤɡ� par enddocument