- 1Some notations:
- 2Polynomial in general context
- 2.1Example 1:
- 2.2Example 2:
- 3Ring of polynomials over
- 4The notation:
- 5Proposition: equality over a function
- 6degree of polynomial
- 7Proposition: on degrees of polynomials
- 8Theorem: Fundamental Theorem of Algebra
- 9Proof of Fundamental Theorem of Algebra
- 9.1Applet devices to understand proof of the Fundamental Theorem of Algebra
- 10Theorem: Remainder Theorem
- 11Corollary: from Remainder Theorem:
- 12Proposition about factorization
- 13Multiplicity of zero
- 14Proposition: Division Algorithm
- 15divisibility of polynomials
- 16Highest common factor (HCF) of polynomials
- 17Lemma about HCF of Polynomials
- 18Euclidean Algorithm for calculating HCF of polynomials
- 19Theorem: w.r.t hcf of two polynomials
- 20polynomial analogue of a prime number
- 21Theorem: w.r.t irreducible polynomials
- 22coprime polynomials
- 23Lemma w.r.t irreducibility and divisibility
- 24Theorem: Uniqueness Theorem
- 25Lemma: Gauss's Lemma
- 26Corollary over Gauss's Lemma
- 27Theorem: Eisenstein's Criterion
- 27.1Example 1:
- 27.2Example 2:
- 28Reduction Modulo p
(This is a work in progess...)
(Note: I am doing some exploration on Galois Theory and I have written these notes on polynomials here for my own convenience. Please write in the comments section if any clarification is required. Also, I will not provide proofs here for the time being.)
The symbol is the domain of complex numbers; i.e. set of all possible complex numbers.
The symbol is the domain of real numbers; i.e. set of all possible real numbers.
The symbol is the domain of rational numbers; i.e set of all possible rational numbers.
The symbol is the domain of integers; i.e set of all possible Integers.
Generally, the symbol is used to denote a subfield of .
Generally, the symbol is used to denote a subring of .
In this blogpost, all subring are assumed to be subring with unity.
divides is denoted as .
does not divide is denoted as .
denotes the degree of polynomial .
The symbol is used to denote element of. For example means is an element of set .
Complex number are of the form , where .
Set of equivalence classes, over integers, is denoted by .
denotes the integers modulo or cyclic group of order .
Under addition, .
Polynomial in general context
A polynomial over in the indeterminate is an expression:
where , and is undefined.
are the coefficients of the polynomial.
is a polynomial over
is a polynomial over
Ring of polynomials over
The set of all polynomials over in the obeys all of the usual algebraic laws.
We denote this set by , and call it the ring of polynomials over in the indeterminate .
We can also define polynomials in several indeterminates , obtaining the ring of -variable polynomials in an analogous way.
An element of will usually be denoted by a single letter, such as , whenever it is clear which indeterminate is involved.
If there is ambiguity, we write to emphasize the role played by
The notation makes appear to be a function, with as its independent variable, and in fact we can identify each polynomial over with the corresponding function.
Each polynomial can be considered as a function from to , defined as follows:
if and , then is mapped to
Proposition: equality over a function
Two polynomials over define the same function if & only if they have the same coefficients.
Proposition implies that we can safely consider a polynomial over a subfield of as either a formal algebraic expression or a function.
Moreover, the same notational flexibility allows us to change the variable in a polynomial.
For example, if are two indeterminates and , then we may define
degree of polynomial
If is a polynomial over and , then the degree of is the highest power of occurring in with nonzero coefficient.
We write for the degree of
Proposition: on degrees of polynomials
If are polynomials over , then
Theorem: Fundamental Theorem of Algebra
Let be a polynomial over , with . Then there exists at least one such that .
In simpler terms: Every complex polynomial has a root.
Proof of Fundamental Theorem of Algebra
Read a proof I found on the internet here: https://www.math.brown.edu/~res/INF/handout5.pdf
Applet devices to understand proof of the Fundamental Theorem of Algebra
Theorem: Remainder Theorem
Let with 1" />, and let
1. There exist and such that,
2. The constant satisfies
Corollary: from Remainder Theorem:
The complex number is a zero of if and only if divides in
Proposition about factorization
Let with . Then there exist , & , such that
Multiplicity of zero
are the only complex zeros of . The zeros need not be distinct.
Collecting together those that are equal, we can rewrite Equation in the form
where are distinct, & are integers and
We call the multiplicity of the zero of
Every complex polynomial of degree has precisely complex zeros, counted according to multiplicity.
Proposition: Division Algorithm
Let & be polynomials over subfield , & suppose that is nonzero. Then there exist unique polynomials and over subfield , such that and has a strictly smaller degree than .
With the above notation, is called the quotient & is called the remainder on dividing by .
The inductive process we employed to find and is called the Division Algorithm.
divisibility of polynomials
Let and be polynomials over subfield . We say that divides (or is a factor of , or is a multiple of ) if there exists some polynomial over such that .
The notation will mean that divides , while will mean that does not divide .
Highest common factor (HCF) of polynomials
A polynomial over field is a highest common factor (HCF) of polynomials and over if and and further, whenever and , we have .
HCF's need not be unique. The next lemma shows that they are unique apart from constant factors.
Lemma about HCF of Polynomials
If is an hcf of the polynomials and over subfield , and if , then is also hcf for & .
If and are two hcf's for and , then there exists a nonzero element such that .
Euclidean Algorithm for calculating HCF of polynomials
Ingredients: Two polynomials and over subfield , both nonzero.
Recipe: For notational convenience let .
Use the Division Algorithm to find successively polynomials and such that
Because the degrees of the decrease, we must eventually reach a point where the process stops; this can happen only if some
The last equation in the list then reads
and it provides the answer we seek.
With the above notation, is an hcf for and .
The above algorithm is the same as long division algorithm that we have always used to find HCF over integers ().
Let , over
We compute an hcf as follows:
Hence is an hcf. So is any rational multiple of it; e.g
Theorem: w.r.t hcf of two polynomials
Let and be nonzero polynomials over subfield , & let be an hcf for and . Then there exist polynomials and over such that:
polynomial analogue of a prime number
A polynomial of
Theorem: w.r.t irreducible polynomials
Any nonzero polynomial over a subring of is a product of irreducible polynomials over .
definition: If and are polynomials over a subfield of with hcf equal to , we say that and are coprime, or is prime to .
The key to unique factorization is a statement analogous to an important property of primes in
Lemma w.r.t irreducibility and divisibility
Let be a subfield of , an irreducible polynomial over , & , polynomials over . If divides , then either divides or divides .
Theorem: Uniqueness Theorem
Lemma: Gauss's Lemma
Let be a polynomial over that is irreducible over . Then , considered as a polynomial over , is also irreducible over
Corollary over Gauss's Lemma
Let and suppose that over there is a factorization into irreducibles:
Then there exist such that and .
is a factorization of into irreducibles in
Theorem: Eisenstein's Criterion
Let, be a polynomial over . Suppose that there is a prime such that
Then is irreducible over
This is irreducible over if and only if
is irreducible over .
Eisenstein's criterion now applies with , showing that is irreducible.
As it stands, is not susceptible to the Eisenstein treatment.
But is obviously irreducible if and only if is irreducible.
Hint: if and only if
If we expand this monster, we obtain
Therefore, Eisenstein's criterion applies with , so is irreducible over .
Reduction Modulo p
The idea is this. There is a natural map in which each maps to its congruence class modulo .
The natural map extends in an obvious way to a map .
Now a reducible polynomial over is a product of polynomials of lower degree, & this factorization is preserved by the map. Provided does not divide the highest coefficient of the given polynomial, the image is reducible over .
So if the image of a polynomial is irreducible over , then the original polynomial must be irreducible over .
But unfortunately, in practice, the trick is to choose the right value for which shows irreducibility.
Also, Since is finite, there are only finitely many possibilities to check when deciding