12 terms

Ring Theory

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(bc)=(ab)c
iii. a*b R for all a,b R
iv. For any a,b,c R, a(b+c)=ab+ac and (ab)c=ac+b*c
A ring F is a field if it is a commutative ring that contains inverses for all nonzero elements in F.
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.
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.
Zero Polynomial
If ai=0 for all i, then f(x)=0 is the zero polynomial.
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.
If there is a z in the field such that f(z)=0, then z is a root.
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
If a subset of a field F is a field in itself, then it is said to be a subfield of F.
Monic Polynomial
A polynomial with the coefficient of the highest power of x as 1 is a monic polynomial.
Cauchy's Theorem
Given a finite group G, if a prime p divides o(G), then G contains an element of order p.