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Abstract algebra
Draft
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England schools
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England university
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Scotland schools
Taxonomy: mathcentre
Taxonomy: Kind of activity
Taxonomy: Context
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History
Newcastle University Mathematics and Statistics 8 years, 11 months ago
Created this.There is only one version of this exam that you have access to.
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1.draftThree parts, where the student has to collect together polynomials in Z3, Z5 and Z7, respectively. The answer to part a has no X term, because they cancel out.
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2.draftExpanding products of 3 linear polynomials over Z3,Z5,Z7
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3.draftGiven polynomial f(X), g(X) over Q, find polynomials q(X) and r(X) over Q such that f(X)=q(X)g(X)+r(X) and degr(X)<degg(X).
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4.draftModular arithmetic. Find the following numbers modulo the given number n. Three examples to do.
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5.Ready to useSolving simple linear equations in Q and Zn for n=13,17 or 19.
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6.Ready to useGiven two numbers, find the gcd, then use Bézout's algorithm to find s and t such that as+bt=gcd(a,b).
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7.draftIn the Gaussian integer ring Z[i] , find the remainder r=r1+r2i, where a>0,b>0 , on dividing a+bi by c+di .
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8.draftIn the ring Z[√2] , find the remainder r=r1+r2√2, where a>0,b>0 , on dividing a+b√2 by c+d√2 .
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9.draftIn the ring Z[√−2] , find the remainder r=r1+r2√−2, where a>0,b>0 , on dividing a+b√−2 by c+d√−2 .
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10.draftf(X) and g(X) are polynomials over Zn. Find their greatest common divisor (GCD) and enter it as a monic polynomial. Hence factorize f(X) into irreducible factors.
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11.draftFactorise 4 polynomials over Z5.
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12.draftf(X) and g(X) as polynomials over the rational numbers Q. Find their greatest common divisor (GCD) and enter as a normalized polynomial.
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