MATH 111C, S 2015, OVERVIEW
Basics
Denitions. Field, subeld, eld homomorphism, prime eld, characteristic.
Theorem 1. For any eld F , either char(F ) = 0 or char(F ) = p, where p is a positive
prime in Z.
Corollary 2. Let F0 be the prime eld of F . Then:
Math 111C, Spring 2015, Problem (9)
(9) Show that the polynomials X 3 2 and X 3 3 in Q[X] do not have isomorphic
splitting elds over Q.
Proof. First note the following: Every polynomial f Q[X] \ Q has a unique splitting eld
(over Q) which is contained in
SUGGESTIONS AND SOME MOTIVATION
FOR REVIEW OF 111B MATERIAL
Prelude, partly doubling up on what I said in class: Whenever we analyze prototypes
of algebraic structures, such as vector spaces, groups, and rings (now elds), we use homomorphisms to compare s
MATH 111C, S 2015, OVERVIEW
Basics
Denitions. Field, subeld, eld homomorphism, prime eld, characteristic.
Theorem 1. For any eld F , either char(F ) = 0 or char(F ) = p, where p is a positive
prime in Z.
Corollary 2. Let F0 be the prime eld of F . Then:
MATH 111C, S 2015, OVERVIEW
Basics
Denitions. Field, subeld, extension eld, eld homomorphism, prime eld, characteristic.
Theorem 1. For any eld F , either char(F ) = 0 or char(F ) = p, where p is a positive
prime in Z.
Corollary 2. Let F0 be the prime eld
KRONECKERS THEOREM, FIRST VERSION
We know: Every embedding : F K of elds induces an embedding of polynomial
rings : F [X] K[X] (i.e., an injective ring homomorphism); it sends any polynomial
d
d
i
i
i=0 ai X F [X] to the polynomial
i=0 (ai )X in K[X].
The
MATH 111C, S 2015, EXISTENCE OF
SPLITTING FIELDS, GENERAL VERSION
Ill start with a reminder. In class, we proved:
Theorem 14 [nite version]. Let F be a eld and P a nite set of nonconstant polynomials in F [X]. Then there exists a splitting eld for P over
MATH 1110 NAME
SECOND MIDTERM
15 May 2015, 10:0010:50 PERM NO.
PLEASE WRITE NEATLY. When giying proofs, be sure to sort out tentative ideas
on scratch paper, and to put down your argument in a logical sequence on the test
paper. The scratch work will not
MATH 1 1 1 C NAME
FIRST MIDTERM
24 April 2015, 10:00-10:50 PERM NO.
PLEASE WRITE NEATLY. When giving proofs, be sure to sort out tentative ideas
on scratch paper, and to put down your argument in a logical sequence on the test
paper. The scratch work Will
Math 111C, Spring 2015, Homework Assignments
All assignments refer to our text, Abstract Algebra, third edition, by Dummit & Foote.
March 30: p.519, # 5
April 1: p.519, # 7, more cleanly stated as follows: Consider the polynomials fn =
x3 nx + 2 Z[x], for
A REMARK ON FINITE ABELIAN GROUPS THAT
FEEDS INTO THE PROOF OF THEOREM 22
Auxiliary Proposition. Let A be a nite abelian group. Then there exists an element
x A such that |y| divides |x| for all y A.
Consequence: If A fails to be cyclic, there exists a na