Review Sheet Math 582

[n] means the topic is found on acetate n.   Acetates after  [83]  are new to the review sheet after the midterm.

Definitions

Integral domain [3]

Examples:  Z   Q   R   C   [4]

Zp where p is prime [4]

D[l], where D  is an integral domain [4]

Euclidean domain [18]

Principle ideal domain  [21]

Field [5] Examples:

Q   R   C   [5]

Zp where p is prime [5]

Quotient field of an integral domain  [6-11]

F(l), where  F  is a field  [11]

Subfield  [22]

Extension field  [22]

Algebraic element in an extension field  [22]

Transcendental element in an extension field  [22]

Minimal polynomial  [27]

Monic polynomial  [52.0]

Degree of an extension field [45.0]

Characteristic of a field  [52.2-56]

Finite fields

Irreducible polynomials over finite fields  [62-64.3]

Construction of finite fields  [65]

Primitive element  [82]

Primitive polynomial  [83]

Ring [10.0]
Homomorphism of rings  [10.1, 12]

An isomorphism of rings is a homomorphism which is  1 - 1 and onto.

Two rings are isomorphic if there is an isomorphism of one onto the other.

Kernel of a ring homomorphism [13]

Ideal [13]

Factor  ring    R/I  [14]

Principle ideal  [21]

Theorems--with proofs If an integral domain is a field, then it is isomorphic to its quotient field.  [10-11]

The kernel of a ring homomorphism is an ideal. [13]

A ring homomorphism is  1-1 if and only if its kernel is {0}. [14]

Every ideal in a Euclidean domain is a principle ideal [19-20]

Every complex number is algebraic over the field of real numbers  [25]

If  E  is an extension field of  F  and  r  is an element of  E  which is algebraic over  F,  then the minimal polynomial for  r  over  F  is unique.  [27-30.1]

The set of all polynomials  in  F[ l ]  which have  r  as a root is the principle ideal generated by the minimal polynomial of  r.  [30.0]

The minimal polynomial of an algebraic element is irreducible.  [30.1]

If  p( l )  is irreducible in   F[ l ]  and is monic and  p(r) = 0,  then   p( l ) is the irreducible polynomial for  r  over  F. [30.1]

If  E  is an extension field of  F  and  r  is an element of  E,  then the mapping  of   F[ l ] into  E given by    (p(l ))f = p(r)  is a homomorphism.  Further, if  r  is transcendental over  F  then the kernel is  {0}.  On the other hand, if  r  is algebraic over  F, then the kernel is the principle ideal generated by the minimal polynomial of  r.  [35-37]

If   E  is an extension field of  F and  r  in  e  is algebraic over  F  and  m(l )  is the minimal polynomial for  r   in  F[ l ], then the factor ring F[ l ]/ (p(l ))  is isomorphic to  F[r], and
the dimension of  F[r]  as a vector space over  F  equals the degree of  m(l ), and  a basis for  F[r]  over  F  is  {1, r, r2, ... rn-1 }. [38, 43-45.0]

If  r  is algebraic over  F,  then every element of  F[r]  is algebraic over  F.  [45.1]

If  E  is an extension field of degree  n  over  F  and  K  is an extension field of degree  m  over  E,  then  K  is an extension field of degree  nm  over  F.  [46-47]

If  F  is a field of characteristic  p, then

(1)  (a + b)p = ap + bp,
(2)  (a - b)p = ap - bp,
(3)  If  q  is a power of  p,  then  (a + b)q = aq + bq, and  (a - b)q = aq - bq,  and (4)  if  aq = bq , then  a = b.  [57-61]
Theorems--no proofs If  f  is a homomorphism of a ring  R  into a ring  S  and f(R)  is the set of images, then   f(R)  and  R/I  are isomorphic,  Where  I   is the kernel of  f. [15-17]

The three classical Greek geometry problems cannot be carried out using straightedge and compass:  squaring the circle, duplicating the cube, trisecting angles  [48-51]

The multiplicative group of nonzero elements  E*  of a finite field  E  is cyclic. [82]

If   is a positive rational number and  a  is a rational number such that  0 < a < 1, then there is a positive integer   N  such that whenever  n  is greater than or equal to  N, we have  an < e.
[102.3-102.4]

Calculations Finding minimal polynomials  [31.1-33, 52.0-52.1]

Determine whether a polynomial is irreducible [31.1-33]

Finding  r-1  when  r  is algebraic over  F. [39]

How to add and multiply in finite fields  [66-68]

Order of a nonzero element under multiplication ion a finite field. [69-70]

Factorization of  l8 - l  in  Z2[ l].  [69-71]

Miscellaneous calculations in finite fields  [72-73]

If  q( l )  is  in   Z[ l ]  and has leading coefficient  a  and  p  is a prime number for which  a  is not  0  mod  p,  and
q*( l )  is  the polynomial in   Zp[ l ] obtained from  q( l ) by reducing all its coefficients mod  p, then  q*( l )  irreducible implies that   q( l ) is irreducible. [98-99]

Geometry

Introduction into the Euclidean plane of points at infinity and the line at infinity  [90]

Use of homogeneous coordinates  [91, 93]

Construction of the real numbers

Definition of cut [100]

Six simple consequences of the definition of cut [101.0-101.1, 102.1&2, 102.5]

Example of the cut  Ca, where  a  is a rational number  [102.0]

Definition of the set of real numbers as the set of all cuts. [106]

Order on  the set  R  of cuts. [106]

Properties of order on cuts  [107-109.1]

Definition of upper bound and least upper bound [109.2]

Least Upper Bound property and its proof [109.2-109.6]

Definition of addition of cuts [110]

Properties of addition  [112-116.3]

Definition and properties of multiplication of cuts [118-130.4]

Notation

F(l) for the quotient field of   F[l]    [11]

R/I  for factor ring  [14]

m( l )  for minimal polynomial  [27]

F[r] for the field generated by an algebraic element  r  over  F. [38]