Computations and processes
Mod a number down (compute x mod m )
Give representatives of a congruence class [ x]m
Determine if a ≡ b(mod m)
Solve linear congruences ax + b ≡ c(mod m) by using cancellation and additive and
multiplicative inverses. Know when there is no solution, and also when you get a union of
congruence classes as the solution.
5. Solve systems of linear congruences using the Chinese Remainder Theorem
6. Perform arithmetic in the Zm rings (including finding additive and multiplicative inverses
of elements)
7. Identify zero divisors and units in the Zm rings
8. Recognize ring, field, and group properties on abstract Cayley tables, locate identity,
inverses, etc.
9. Perform matrix addition and multiplication, verify zero divisors, inverses for matrices
10. Compute the characteristic of a ring
11. Computations with complex numbers (modulus, conjugate, simplifying expressions,
dividing by rationalizing, finding multiplicative inverse)
12. Converting complex numbers between forms (rectangular/polar)
13. Using deMoivre's Formula to find roots of complex numbers, including knowing what
"roots of unity" is asking for.
14. PreCalc level polynomial algebra (add, multiply, long division, synthetic division)
15. Polynomial algebra in the Zm [ x] rings.
16. Writing polynomials as g ( x) = p( x)q ( x) + r ( x) using the Division Algorithm
17. Using Euclid's Algorithm to get the gcd of polynomials. Know what "monic gcd" is
asking for.
18. Reverse Euclid's Algorithm to express gcd of polynomials as d ( x ) = s ( x) f ( x ) + t ( x ) g ( x )
19. Apply Root/Remainder theorem to test for factors. Combine with long division/synthetic
division to factor polynomials.
20. Completely factor a polynomial by throwing roots at it. Know how factorizations vary in
][ x], _[ x], \[ x], ^[ x] , and how to do this in Zm [ x] . And don't forget the quadratic
formula.
21. Calculate content of a polynomial (and state if primitive)
22. Apply Eisenstein's criteria to determine irreducible
23. Mod polynomial down into Zp [ x] to determine irreducible
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2.
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4.
24. Find real and imaginary parts of a complex valued function
25. Computations in various group examples ( < Zm , + >, < U m ,* > , dihedral groups, Klein 4group) using Cayley tables to get results of operations, find inverses, solve equations, etc.)
26. Generate cyclic subgroups by cycling through powers (or multiples, if additive group) of
an element. Identify order of an element in a group
27. Perform operations in direct product groups