The polynomial ring F[x]

Question 1

Degree of a polynomial

Answer 1

The degree of f = a_nxⁿ +…+a₀ is

deg f = { m, if f =/= 0 and m is the coefficient of the highest order term; or

-∞ if f = 0.

Question 2

Irreducible

Answer 2

f is irreducible if f = gh ( g,h in F[x]) implies g or h has degree zero.

Question 3

Monic

Answer 3

f is monic if a_n = 1.

Question 4

gcd

Answer 4

d is the gcd of f,g in F[x] if

• d in F[x] is monic,
• d | f and d | g,
• if c | f and c | g then c | d.

Question 5

Root

Answer 5

An element a in F is a root of f in F[x] if f(a) = 0.

Question 6

1. Division with remainder. For f, 0 =/= g in F[x], there exists r,s in F[x] st f = gr + s and deg s < deg g.
2. F[x] is a PID (and thus a UFD), ie. f = uf₁…f_n for all f, where u has degree 0 and f_i are monic irreducible polynomials.
3. Suppose d =gcd(f,g) in F[x]. Then there exist polynomials r,s in F[x] st d = fs + gr.

Question 7

Let f be in F[x]. An element a in F is a root of f iff x-a divides f.

Answer 7

There exists g,r in F[x] st f = (x-a)g + r and

deg r < deg (x-a) = 1.

f(a) = 0 ⇔ f(a) = (a-a)g(a) +r(a) = 0

⇔ r = 0 as r is constant

⇔ x-a | f.

Question 8

1. • and • are well defined,
2. ( R/I, +, • ) is a ring with [1] the multiplicative identity and [0] the additive identity.