FINAL!!!

Question 1

Leading Coefficient/Monic Polynomial

Answer 1

Let f(x)=a0+ ∑ aix^i ∈ R[x]. If an /= 0R, then an is called the leading coefficient of f (x) and n is the degree of f (x), denoted deg f (x). If an = 1R /= 0R, then f (x) is called a monic polynomial.

Question 2

The Division Algorithm in F[x]

Answer 2

Let F be a field and f(x), g(x) ∈ F[x] with g(x) =/ 0F . Then there exist unique polynomials q(x), r(x) ∈ F[x] such that f(x) = g(x)q(x)+r(x) and either r(x) = 0F or deg r(x) < deg g(x)

Question 3

Divisor

Answer 3

Let f(x), g(x) ∈ F[x], where F is a field and g(x) =/ 0F . Then g(x) is a divisor of f(x) or g(x) is said to divide f(x) in F[x], written g(x) I f(x), if f(x) = g(x)h(x) for some h(x) ∈ F[x].

Question 4

Greatest Common Divisor

Answer 4

Let f(x), g(x) ∈ F[x], both not equal to 0F . The greatest common divisor (gcd) of f(x) and g(x) is the monic polynomial d(x) of highest degree that divides both f(x) and g(x).

Thus d(x) = gcd (f(x), g(x)) iff
- d(x) is monic,
- d(x) I f(x) and d(x) I g(x),
- c(x) I f(x) and c(x) I g(x) implies deg c(x) < deg d(x)

Question 5

Relatively Prime

Answer 5

If gcd (f(x), g(x)) = 1F , then f(x) and g(x) are relatively prime in F[x].

Question 6

Associate

Answer 6

Let R be a commutative ring with identity and a, b ∈ R. Then a is an associate of b if a = bu for some unit u in R.

Question 7

Irreducible/Reducible

Answer 7

Let F be a field. A nonconstant polynomial p(x) ∈ F[x] is said to be irreducible if its only divisors in F[x] are its associates and the units of F[x]. A nonconstant polynomial that is not irreducible is said to be reducible.

Question 8

Polynomial Function

Answer 8

Let R be a commutative ring and let f(x)=a0+ ∑ aix^i ∈ R[x]. Then there is a function f : R -> R whose rule of assignment is f(r) = a0+ ∑ air^i ∈ R[x]. The function f is said to be induced by the polynomial f(x) and is called a polynomial function.

Question 9

Root

Answer 9

Let f(x) ∈ R[x], where R is a commutative ring and let a ∈ R. Then a is a root of f(x) if f(a) = 0R.

Question 10

The Remainder Theorem

Answer 10

Let F be a field, f(x) ∈ F[x] and a ∈ F. Then there exists a unique polynomial q(x) ∈s F[x] such that f(x) = (x - a)q(x) + f(a).

Question 11

The Factor Theorem

Answer 11

Let F be a field, f(x) ∈ F[x] and a ∈ F. Then a is a root of f(x) iff (x - a) I f(x).

Question 12

Congruence

Answer 12

Let F be a field and f(x), g(x), p(x) ∈ F[x]. Assume p(x) =/ 0F . Then f(x) is congruent to g(x) modulo p(x), written f(x) ≡ g(x) (mod p(x)), provided that p(x) I f(x) - g(x).

Question 13

Congruence Class

Answer 13

Let F be a field and f(x), p(x) ∈ F[x] with p(x) =/ 0F . The congruence class of f(x) modulo p(x), denoted [f(x)], is the set of all polynomials in F[x] that are congruent to f(x) modulo p(x).

Question 14

Congruence Class Arithmetic

Answer 14

Let p(x) ∈ F[x] with deg p(x) > 0. Addition and multiplication in F[x]=(p(x)) are defined by...
[f(x)] + [g(x)] = [f(x) + g(x)] and [f(x)][g(x)] = [f(x)g(x)]: