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Modern Algebra Review

Modern Algebra Review. Sections 21, 22, and 23 Josée van den Hoogen and Gabrielle Vasey April 11 th , 2013. Definitions. Group < G, * > Closed under * * is associative There is an identity Every element has an inverse. Definitions. Ring < R, +, ∘ >

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Modern Algebra Review

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  1. Modern Algebra Review Sections 21, 22, and 23 Josée van den Hoogen and Gabrielle Vasey April 11th, 2013

  2. Definitions • Group < G, * > • Closed under * • * is associative • There is an identity • Every element has an inverse

  3. Definitions • Ring < R, +, ∘ > • Closed under addition and multiplication • < R, + > is abelian (+ is commutative) • Multiplication is associative • Distributive laws hold • A commutative ring is a ring where multiplication is commutative

  4. Definitions • Divisors of zero • A divisor of 0 is some element a ≠ 0 such that ∃ b ≠ 0 and ab = 0

  5. Definitions • Integral Domain • A commutative ring with unity 1 ≠ 0 • No divisors of zero

  6. Definitions • Unity • A ring has unity if there is a multiplicative identity • If it exists, the identity is denoted by 1 • Elements that have multiplicative inverses are units

  7. Definitions • Field • A commutative ring with unity and all non-zero elements are units

  8. Section 21 The Field of Quotients of an Integral Domain

  9. Field of Quotients • Theorem • Any integral domain D can be embedded in a field F such that every element of F can be expressed as a quotient of two elements from D • Standard example:in • = { p/q ⎮ p, q ∈ and q ≠ 0 }

  10. Equivalence Relation • (a,b) ∼ (c,d) meaning ad=bc • Addition: [(a,b)] + [(c,d)] = [(ad+bc, bd)] • Identity: [(1,0)] • Inverse for [(a,b)] is [(-a,b)] • Multiplication: [(a,b)] * [(c,d)] = [(ac, bd)] • Identity: [(1,1)] • Inverse for [(a,b)] is [(b,a)]

  11. Uniqueness • Every field L containing an integral domain D must contain the field of quotients of D • A field of quotients is minimal

  12. Section 22 Rings of Polynomials

  13. Polynomials f(x) Let be a ring. A polynomial f(x) with coefficients in R is an infinite formal sum where and for all but a finite number of values of i.

  14. Degree • The largest value isuch that ai≠ 0 and aj= 0 for j > iis the degree of the polynomial. • If only a0 ≠ 0, then you just have an element from the ring itself. This is a constant polynomial with degree 0. • If all ai = 0, then we have the zero polynomial and the degree is undefined.

  15. Theorem • R[x], the set of all polynomials with coefficients from R is a ring. If R is commutative, so is R[x]. If R has unity 1 ≠ 0, then 1 is also unity for R[x]. • R[x,y] set of polynomials with two intermediates, can also be written as (R[x])[y].

  16. Evaluation Homomorphism • Let F be a subfield of a field E, let α be any element of E, and let x be an indeterminate. The map ϕα: F[x] → E defined by: • ϕα(a0 + a1x + …+ anxn) = a0+ a1α + …+ anαn • for (a0 + a1x + … + anxn) ∈ F[x] is a homomorphism of F[x] into E.

  17. Section 23 Factorization of Polynomials over a Field

  18. The Division Algorithm Let f(x) = anxn + an-1xn-1 + …+ a0 where g(x) = bmxm + bm-1xm-1 + … + b0 be two elements of F[x], with an and bm both nonzero elements of F and m > 0. Then there are unique polynomials q(x) and r(x) in F[x] such that f(x) = g(x)q(x) + r(x), where either r(x) = 0 or the degree of r(x) is less than the degree m of g(x).

  19. Factor Theorem • An element a ∈ F is a zero of f(x) ∈ F[x] if and only if x – a is a factor of f(x) in F[x]. • Corollary: • A nonzero polynomial f(x) ∈ F[x] of degree n can have at most n zeros in a field F.

  20. Irreducible Polynomials • A non-constant polynomial f(x) ∈ F[x] is irreducible over F if f(x) cannot be expressed as a product g(x)h(x) of two polynomials g(x) and h(x) in F[x] both of lower degree than the degree of f(x). Otherwise we say it is reducible.

  21. Theorems • Let f(x) ∈ F[x], let f(x) be of degree 2 or 3. Then f(x) is reducible over F if and only if it has a zero in F. • f(x) factors in [x] if and only if it factors over [x], and the degrees of the factors are the same.

  22. Corollary • If f(x) = xn + an-1xn-1 + … + a0 is in [x] with a0 ≠ 0, and if f(x) has a zero in , then it has a zero m in , and m must divide a0.

  23. Eisenstein Criterion • Let p ∈ be a prime. Suppose that f(x) = anxn + … + a0is in [x], and an is not congruent to 0 mod p, but ai = 0 mod p for all i < n, with a0 not congruent to 0 mod p2. Then f(x) is irreducible over .

  24. pthcyclotonic polynomial is irreducible over for any prime p. ie x4 + x3 + x2 + x is irreducible is irreducible if and only if is irreducible.

  25. Factorization • Let p(x) be irreducible in F[x]. If p(x) divides r(x)s(x) then p(x) divides r(x) or s(x). • If p(x) divides r1(x)r2(x)…ri(x) then p(x)divides at least one rj(x)

  26. Uniqueness of Factorization • If F is a field, then every non-constant polynomial f(x) in F[x] can be factored in F[x] into a product of irreducible polynomials. This irreducible polynomial is unique except for order and for unit factors in F.

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