5 files changed, 125 insertions, 0 deletions
diff --git a/3293/CH4/EX4.10/Ex4_10.sce b/3293/CH4/EX4.10/Ex4_10.sce
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+//page 135
+//Example 4.10
+clc;
+clear;
+close;
+x = poly(0,"x");
+P = x^2 + 1;
+disp(P,'P = ');
+disp('P is reducible over complex numbers as: ');
+disp('=',P);
+disp('(x-i)(x+i)');
+disp('Whereas, P is irreducible over real numbers as:.');
+disp('=',P);
+disp('(ax + b)(a''x + b'')');
+disp('For, a,a'',b,b'' to be in R,');
+disp('aa'' = 1');
+disp('ab'' + ba'' = 0');
+disp('bb'' = 1');
+disp('=> a^2 + b^2 = 0');
+disp('=> a = b = 0');
+//end
diff --git a/3293/CH4/EX4.3/Ex4_3.sce b/3293/CH4/EX4.3/Ex4_3.sce
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index 000000000..9d1eb2b16
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+++ b/3293/CH4/EX4.3/Ex4_3.sce
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+//page 121
+//Example 4.3
+clc;
+clear;
+close;
+disp('C is the field of complex numbers');
+x = poly(0,"x");
+f = x^2 + 2;
+disp(f,'f = ');
+//part a
+disp('if a = C and z belongs to C, then f(z) = z^2 + 2');
+disp(horner(f,2),'f(2) = ');
+disp(horner(f,(1+%i)/(1-%i)),'f(1+%i/1-%i) = ');
+disp('----------------------------------------');
+//part b
+disp('If a is the algebra of all 2*2 matrices over C and');
+B = [1 0;-1 2];
+disp(B,'B = ');
+disp(2*eye(2,2) + B^2,'then, f(B) = ');
+disp('----------------------------------------');
+//part c
+disp('If a is algebra of all linear operators on C^3');
+disp('And T is element of a as:');
+disp('T(c1,c2,c3) = (i*2^1/2*c1,c2,i*2^1/2*c3)');
+disp('Then, f(T)(c1,c2,c3) = (0,3*c2,0)');
+disp('----------------------------------------');
+//part d
+disp('If a is the algebra of all polynomials over C');
+g = x^4 + 3*%i;
+disp(g,'And, g = ');
+disp(horner(f,g),'Then f(g) = ');
+//end
diff --git a/3293/CH4/EX4.7/Ex4_7.sce b/3293/CH4/EX4.7/Ex4_7.sce
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+//page 131
+//Example 4.7
+clc;
+clear;
+close;
+x = poly(0,"x");
+p1 = x + 2;
+p2 = x^2 + 8*x + 16;
+disp('M = (x+2)F[x] + (x^2 + 8x + 16)F[x]');
+disp('We assert, M = F[x]');
+disp('M contains:');
+t = p2 - x*p1;
+disp(t);
+disp('And hence M contains:');
+disp(t - 6*p1);
+disp('Thus the scalar polynomial 1 belongs to M as well all its multiples.') 
+//end
diff --git a/3293/CH4/EX4.8/Ex4_8.sce b/3293/CH4/EX4.8/Ex4_8.sce
new file mode 100755
index 000000000..97cdcbf2a
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+//page 133
+//Example 4.8
+clc;
+clear;
+close;
+x = poly(0,"x");
+//part a
+p1 = x + 2;
+p2 = x^2 + 8*x + 16;
+disp(p1,'p1 = ');
+disp(p2,'p2 = ');
+disp('M = (x+2)F[x] + (x^2 + 8x + 16)F[x]');
+disp('We assert, M = F[x]');
+disp('M contains:');
+t = p2 - x*p1;
+disp(t);
+disp('And hence M contains:');
+disp(t - 6*p1);
+disp('Thus the scalar polynomial 1 belongs to M as well all its multiples');
+disp('So, gcd(p1,p2) = 1');
+disp('----------------------------------------------');
+//part b
+p1 = (x - 2)^2*(x+%i);
+p2 = (x-2)*(x^2 + 1);
+disp(p1,'p1 = ');
+disp(p2,'p2 = ');
+disp('M = (x - 2)^2*(x+%i)F[x] + (x-2)*(x^2 + 1');
+disp('The ideal M contains p1 - p2 i.e.,');
+disp(p1 - p2);
+disp('Hence it contains (x-2)(x+i), which is monic and divides both,');
+disp('So, gcd(p1,p2) = (x-2)(x+i)');
+disp('----------------------------------------------');
+//end
diff --git a/3293/CH4/EX4.9/Ex4_9.sce b/3293/CH4/EX4.9/Ex4_9.sce
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index 000000000..569b30889
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+//page 133
+//Example 4.9
+clc;
+clear;
+close;
+disp('M is the ideal in F[x] generated by:');
+disp('(x-1)*(x+2)^2');
+disp('(x+2)^2*(x+3)');
+disp('(x-3)','and');
+x = poly(0,"x");
+p1 = (x-1)*(x+2)^2;
+p2 = (x+2)^2*(x-3);
+p3 = (x-3);
+disp('M = (x-1)*(x+2)^2 F[x] + (x+2)^2*(x-3) + (x-3)');
+disp('Then M contains:');
+t = 1/2*(x+2)^2*((x-1) - (x-3));
+disp(t);
+disp('i.e., M contains (x+2)^2');
+disp('and since, (x+2)^2 = (x-3)(x-7) - 17');
+disp('So M contains the scalar polynomial 1.');
+disp('So, M = F[x] and given polynomials are relatively prime.');
+//end