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.

proofs to come soon...

My internet has been down at my house for a bit now, but you can expect some cool proofs to come soon. Look for a proof to Gauss's Lemma -- it's pretty sexy.
Carry on.