### Ring

Let R be a set with binary operations denoted + and ** and referred to as addition and multiplication. Then <R,+,**> is called a ring if the following properties hold

i. (R,+) is an abelian group

ii. For any a,b,c in R a**(b**c)=(a**b)**c

iii. a*b R for all a,b R

iv. For any a,b,c R, a**(b+c)=a**b+a**c and (a**b)**c=a**c+b*c

### Field

A ring F is a field if it is a commutative ring that contains inverses for all nonzero elements in F.

### Characteristic

The smallest positive integer n such that nr=0 for any element r in the ring R is called the characteristic. That additive order of any r divides n. If no such n exists, then we say the ring has characteristic 0.

### Polynomial

Let R be a commutative ring and let X be an indeterminate(same symbol not in R). A polynomial in X with coefficients from R is anything of the form sum aixi where ai R and all but a finite number of ai are zero.

### Degree

The largest index d for which ad is not equal to zero and ai=0 for i>d. We define degree(f)=d unless f is the zero polynomial and in that case f does not have a degree.

### Zero Divisor

Let R be a ring and a in R**. If there exists a b in R** s.t. ab=0 or ba=0, then a is called a zero divisor.

### LaGrange's Interpolation Formula

Let F be a field and n>0 some integer. For any selection of n+1 distinct elements a0,a1,...an and any selection of n+1 arbitrary elements b0,b1,...,bn, there exists a unique polynomial f in F[x] of dgree at most n which satisfies f(ai)=bi for i=0,1,...,n

### Monic Polynomial

A polynomial with the coefficient of the highest power of x as 1 is a monic polynomial.