Study Guide 1 for Ring and Field Theory, Midterm 2

So it's midterm time again. Here are some definitions I'll need to iterate on the exam. Later, I might put up some proofs. Honestly, I've found the material this term much easier than last term, so I haven't felt the need to blog as much. Sorry.

• Principal Ideal Domain (PID): An integral domain D is a PID if every ideal is of the form 'a'={ar| r in D}.
• Ring of Polynomials: Let R be a commutative ring. The set of formal symbols R[x]={a(n)x^n+a(n-1)x^(n-1)+...+a(1)x+a(0)| a(i) in R, n is a nonnegative integer} is called the ring of polynomials over R in the indeterminate x. Two elements, a(n)x^n+a(n-1)x^(n-1)+...+a(1)x+a(0) and b(m)x^m+b(m-1)x^(m-1)+...+b(1)x+b(0), are said to be equal iff a(i)=b(i) for all nonnegative integers i. Further, define a(i)=0 when i>n and b(i)=0 when i>m.
• Content of a Polynomial, Primitive Polynomial: In a polynomial ring R[x], the content of the element a(n)x^n+a(n-1)x^(n-1)+...+a(1)x+a(0) is the greatest common divisor of the coefficients, a(i). The element a(n)x^n+a(n-1)x^(n-1)+...+a(1)x+a(0) is said to be primitive if the content is 1 (that is, if at least one coefficient is relatively prime to all the others).
  
• Irreducible Polynomial: Let D be an integral domain. A polynomial f(x) from D[x] that is neither the zero polynomial nor a unit is said to be irreducible if, when f(x)=g(x)h(x), where g(x) and h(x) are in D[x], g(x) or h(x) is a unit in D[x].
  
• Prime Element: Let D be an integral domain, and let a, b, c be in D. The nonzero, non-unit element a is prime if, if a|bc then a|b or a|c.
  
• Vector Space: A set V is said to be a vector space over a field F is V is an Abelian group under addition, and, if for each a in F and v in V, there is an element av in V such that the following conditions hold for all a,b in F and all u,v in V: 1) a(v+u)=av+au; 2) (a+b)v=av+bv; 3) a(bv)=(ab)v; 4) 1v=v.
  
• Linear Independence: A set of vectors S is said to be linearly independent over a field F if there are vectors v1, v2,..., vn from S and elements a1, a2,...,an from F, not all zero, such that a1v1+a2v2+...+anvn=0.
  

Ok, those are the definitions I need to know. Pretty straightforward. The vector space definition is a little long winded, as is the polynomial ring definition, but I'll have to make do... Until proof time, folks.