LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Non-Nested Real Algebraic Numbers By MapleKahtan H. Alzubaidy Key Words : Algebraic Number, Conjugates , Minimal Polynomial Theoritical Background Number Fields \316\261\342\210\210\342\204\202 is called an algebraic number , if there is f(x)\342\210\210\342\204\232[x] such that f(\316\261) = 0. If f(x) is monic and of smallest degree n , we say that \316\261 is an algebraic number of degree n and f(x) is the minimal polynomial of \316\261 . Note that the minimal polynomial is irreducible over \342\204\232 . We are concerned only with nonnested algebraic numbers of degrees \342\211\244 7.The number field induced by \316\261 is given by \342\204\232(\316\261) = {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\342\204\232(\316\261) is n-dimensional vector space over \342\204\232 with basis {1,\316\261,..,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 \316\261 and \316\262 are two algebraic numbers of derees m and n respectively, We define \342\204\232(\316\261, \316\262) = \342\204\232(\316\261)(\316\262). \342\204\232(\316\261,\316\262) is an assciative algebra over \342\204\232 of dimension mn.Algebraic operations addition, subtraction, multiplication and division on algebraic numbers can be done very well by using maple command (evala) .Examples1) \342\204\232(LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictSSZtc3FydEdGJDYjLUYjNiYtSSNtbkdGJDYmUSI1RicvJSVib2xkR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSVib2xkRicvJStmb250d2VpZ2h0R0Y9RjhGO0Y+RjhGO0Y+) = { 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2) \342\204\232(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3) \342\204\232(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) = {a+ 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 1,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4) \342\204\232(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 Representations Theorem 1 Let \316\261 be an algebraic number of degree n . Then there is an algebra monomorphism from \342\204\232(\316\261) into 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 \316\261 and \316\262 are two algebraic numbers of derees m and n respectively, we get a monomorphism from \342\204\232(\316\261, \316\262) into LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1YkdGJDYlLUkjbWlHRiQ2J1EiTUYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGNC8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictRiM2KS1GLzYnUSNtbkYnRjJGNUY3RjpGMkY1LyUrZm9yZWdyb3VuZEdRLFsyMDAsMCwyMDBdRicvJSxwbGFjZWhvbGRlckdGNEY3RjovJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1GLzYjUSFGJ0YyL0Y4RjxGOg==(\342\204\232) .So an algebraic number in a number field is represented by a square matrix by using the idea of linear transformation and basis.The matrix corresponding to the algebraic number 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 Examples1) In \342\204\232( 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) In \342\204\232(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a+ 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 Let 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 Note that a field homomorphism is always one to one. Let E be a number field . Then a 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Theorem 2 A number field of degree n has n distinct embeddings into \342\204\202. Definition. Let E:\342\204\232 be a field extension of degree n and \316\261\342\210\210E. The conjugates of \316\261 are 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 The minimal polynomial of \316\261 is given by 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 Trace and norm of \316\261 are defined by trace(\316\261) = 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 Examples 1) \342\204\232(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 Let \316\270: \342\204\232(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 \316\270(s)=s . Then \316\270((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 These determine the n embeddings ( conjugates) .2) \342\204\232 (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 If \316\270: \342\204\232(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 Then \316\270 : (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(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(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(- 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(- 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJStleGVjdXRhYmxlR0Y9Rjk=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 Matrix Representations The mattrix representation of the algedraic 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 B ( 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JSFH Special MatrixM:=proc()local n,L,h,K;n:=nargs;L:=[seq(args[i],i=1..n)];h := X -> Rotate(X,n-1);K:=[L,seq((h@@r)(L),r=1..n-1)];Transpose(Matrix(n,n,K));end proc;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B:=proc() local m,SS,t,A,A1,A2,A3;m:=nargs;SS:=seq(args[i],i=1..m-1);t:=args[m];A:=M(SS);A1:=t*Matrix(m-1, A, shape = triangular[upper]);A2:=t*Matrix(m-1, A, shape = [diagonal]);A3:=Matrix(m-1,A, shape = triangular[lower]);(A1-A2)+A3;end proc;LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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 Matrix representations of the algebraic nubers 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 B22 means 2-block extended by 2-block and B32 means 3-block extended by 2-block.B22:=proc(a,b,c,d,s,t);blockmatrix(2, 2, [B(a, b, s), t*B(c, d, s), B(c, d, s), B(a, b, s)]);end proc;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JSFHB32:=proc(a,b,c,d,e,f,s,t);blockmatrix(2, 2, [B(a, b, c, s), t*B(d, e, f, s), B(d, e, f, s), B(a, b, c, s)]);end proc;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JSFH Conjugates The conjugates of of 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 The procedure (an) generates the algebraic number 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 .an:=proc() local n,L,s;n:=nargs;L:=[seq(args[i],i=1..n)];s:=args[n];args[1]+sum(args[r]*s^((r-1)/(n-1)),r=2..n-1);end proc;LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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JSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= The procedure (cn) computes the complex cojugates cn:=proc()local n,s,L,LL,K,MM;n:=nargs;L:=[seq(args[i],i=1..n)];s:=args[n];K:=seq(w^r,r=1..n-2);LL:=[seq(L[m]*s^((m-1)/(n-1)),m=2..n-1)];LL:=Matrix(1,n-2,LL);MM:=multiply(LL,M(K));seq(print(args[1]+MM[1,u]),u=1..n-2),print(w^(n-1)=1);end proc;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:=proc()local S;S:=seq(args[i],i=1..nargs);print("conjugates of ", an(S), "are"), cn(S);end proc;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JSFHJSFH The onjugates of 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:=proc(a,b,c,d,s,t)local z,S,L,LL;z := (1+sqrt(t))*(1+sqrt(s));z := expand(z);S := [op(z)];L := [a, b, c, d];LL:=[seq(L[r]*S[r], r = 1 .. 4)];add(u, `in`(u, LL));end proc;LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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:=proc(a,b,c,d,s,t);print("conjugates of ", an22(a,b,c,d,s,t), "are"),print(an22(a, -b, c, -d, s, t)),print( an22(a, b, -c, -d, s, t)),print(an22(a, -b, -c, d, s, t));end proc;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JSFH The conjugates of 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:=proc(a,b,c,d,e,f,s,t)local z,S,L,LL;z := (1+sqrt(t))*(1+(s)^(1/3) +(s)^(2/3));z := expand(z);S := [op(z)];L := [a, b, c, d,e,f];LL:=[seq(L[r]*S[r], r = 1 .. 6)];add(u, `in`(u, LL));end proc;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conj32:=proc(a,b,c,d,e,f,s,t)local LK;LK:=[an32(a, b, c, -d, -e, -f, s, t),an32(a, b*w, c*w^2, d, e, f, s, t),an32(a, b*w, c*w^2, -d, -e, -f, s, t),an32(a, b*w^2, c*w, d, e, f, s, t),an32(a, b*w^2, c*w, -d, -e, -f, s, t)];print("conjugates of ", an32(a,b,c,d,e,f,s,t), "are"),seq(print(u),u in LK),print(w^3=1);end proc;LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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JSFHJSFH Minimal polynomial mp(x) and characteristic polynomial ch(x) of an algebraic number mp(x) | ch(x) and mp(x) = ch(x) if ch(x) is irreducible or deg(mp(x)) = deg(ch(x)). In our case mp(x) = ch(x) . Minimal polynomial of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZTLUklbXN1YkdGJDYlLUkjbWlHRiQ2J1EiYUYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dGNC8lLG1hdGh2YXJpYW50R1EsYm9sZC1pdGFsaWNGJy8lK2ZvbnR3ZWlnaHRHUSVib2xkRictRiM2Ki1JI21uR0YkNiZRIjFGJ0YyL0Y4RjxGOkYyRjUvJStmb3JlZ3JvdW5kR1EsWzIwMCwwLDIwMF1GJy8lLHBsYWNlaG9sZGVyR0Y0LyUwZm9udF9zdHlsZV9uYW1lR1ElVGV4dEYnRjdGOi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2L1EiK0YnRjJGQ0Y6LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZVLyUpc3RyZXRjaHlHRlUvJSpzeW1tZXRyaWNHRlUvJShsYXJnZW9wR0ZVLyUubW92YWJsZWxpbWl0c0dGVS8lJ2FjY2VudEdGVS8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRl5vLUYsNiVGLi1GIzYqLUZANiZRIjJGJ0YyRkNGOkYyRjVGREZHRklGN0Y6RkwtRi82I1EhRictSSZtcm9vdEdGJDYkLUYjNistRi82J1Eic0YnRjJGNUY3RjpGMkY1L0ZFUStbMCwxNjAsODBdRidGRy8lNnNlbGVjdGlvbi1wbGFjZWhvbGRlckdGNEZJRjdGOi1GIzYqLUYvNidRIm5GJ0YyRjVGN0Y6RjJGNUZERkdGSUY3RjpGTy1GLDYlRi4tRiM2Ki1GQDYmUSIzRidGMkZDRjpGMkY1RkRGR0ZJRjdGOkZMRmhvLUZccDYkLUYjNistSSVtc3VwR0YkNiVGYHBGY28vJTFzdXBlcnNjcmlwdHNoaWZ0R0ZORjJGNUZjcEZHRmVwRklGN0Y6RmdwRk8tRlA2L1EjLi5GJ0YyRkNGOkZTRlZGWEZaRmZuRmhuRmpuRlxvL0Zgb1EmMC4wZW1GJy1GUDYvUSIuRidGMkZDRjpGU0ZWRlhGWkZmbkZobkZqbi9GXW9GYHJGX3JGTy1GLDYlRi5GZ3BGTEZoby1GXHA2JC1GIzYsLUYvNiVGYnBGNS9GOFEnaXRhbGljRictRmhxNiVGaG8tRiM2LEZpcC1GUDYvUSgmbWludXM7RidGMkZDRjpGU0ZWRlhGWkZmbkZobkZqbkZcb0Zfb0Y/RjJGNUZERkdGSUY3RjpGanFGMkY1RmNwRkdGZXBGSUY3RjpGZ3AtRlA2L1EifkYnRjJGQ0Y6RlNGVkZYRlpGZm5GaG5Gam5GZHJGX3JGZnNGZnNGZnMtRlA2L1EjaW5GJ0YyRkNGOkZTRlZGWEZaRmZuRmhuRmpuRmRyRl9yRmZzRmZzLUYvNidRKCYjODQ3NDtGJ0YyL0Y2RlVGQ0Y6LUkobWZlbmNlZEdGJDYmLUYjNilGaG9GW3BGaG9GMkZJRkNGOkYyRkNGOkZmcy1GLzYnUSNpc0YnRjJGNUY3RjpGZnMtRi82J1EmZ2l2ZW5GJ0YyRjVGN0Y6RmZzLUZQNi9RI2J5RidGMkZDRjpGU0ZWRlhGWkZmbkZobkZqbkZkckZfci1GUDYtRmhzL0Y4USdub3JtYWxGJ0ZTRlZGWEZaRmZuRmhuRmpuRmRyRl9yLUYvNidRJHRoZUYnRjJGX3RGQ0Y6RmZzLUYvNidRKmZvbGxvd2luZ0YnRjJGX3RGQ0Y6RmZzLUYvNidRKnByb2NlZHVyZUYnRjJGX3RGQ0Y6RmZzRmZzLUYvNidRI21wRidGMkZfdEZDRjotRmF0NiYtRiM2J0Zob0YyRklGQ0Y6RjJGQ0Y6RmZzRmZzRmZzLUZQNi1GY3JGYHVGU0ZWRlhGWkZmbkZobkZqbkZkckZfckYyRklGQ0Y6LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=mp:=proc()local SS;SS:=seq(args[i],i=1..nargs);CharacteristicPolynomial(B(SS),x);end proc;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 Minimal polynomials of 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 and 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mp22:=proc(a,b,c,d,s,t); B22(a,b,c,d,s,t);collect(charpoly(B22(a,b,c,d,s,t),x),x);end proc;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mp32:=proc(a,b,c,d,e,f,s,t); B32(a,b,c,d,e,f,s,t);sort(collect(charpoly(B32(a,b,c,d,e,f,s,t),x),x),x);end proc;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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEmZXZhbGFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2Iy1GLDYlUSIlRidGL0YyL0YzUSdub3JtYWxGJy1JI21vR0YkNi1RIjtGJ0Y9LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjI3Nzc3NzhlbUYnTrace and Norm 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tr:=proc()local S; S:=seq(args[i],i=1..nargs); trace(B(S)); end proc;LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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nr:=proc() local S;S:=seq(args[i],i=1..nargs);det(B(S));end proc;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JSFHtr22:=proc(a,b,c,d,s,t);trace(B22(a, b, c, d, s, t));end proc;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:=proc(a,b,c,d,s,t);det(B22(a, b, c, d, s, t));end proc;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:=proc(a,b,c,d,e,f,s,t);trace(B32(a, b, c, d,e,f, s, t));end proc;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JSFHnr32:=proc(a,b,c,d,e,f,s,t);det(B32(a, b, c, d,e,f, s, t));end proc;LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= References 1) T.W.Hungerford, Algebra, Springer, New York, 1974. 2) N. Jacobson, Basic Algebra, 2nd edition, Freeman, New York, 1985. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=TTdSMApJNVJUQUJMRV9TQVZFLzM5MTEyMTUyWCwlKWFueXRoaW5nRzYiNiJbZ2whIiUhISEjKiIkIiQlImFHJSJiRyUiY0dGKUYnRihGKEYpRidGJQ==TTdSMApJNlJUQUJMRV9TQVZFLzEzOTAxMDg5MlgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIyUiIyIjJSJhRyUiYkcqJiUic0ciIiJGKEYrRidGJQ==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