Definition and Theorems (Chapter 4)

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Definition. Let F be a field. A linear algebra over the field F is a vector space \Large{\frak a} over F with an additional operation called multiplication of vectors which associates with each pair of vectors \alpha, \beta in \Large{\frak a} a vector \alpha\beta in \Large{\frak a} called the product of \alpha and \beta in such a way that
( a ) multiplication is associative,

\displaystyle{\alpha(\beta\gamma)=(\alpha\beta)\gamma}

( b ) multiplication is distributive with respect to addition,

\displaystyle{\alpha(\beta+\gamma)=\alpha\beta+\alpha\gamma\quad\text{and}\quad(\alpha+\beta)\gamma=\alpha\gamma+\beta\gamma}

( c ) for each scalar c in F,

\displaystyle{c(\alpha\beta)=(c\alpha)\beta=\alpha(c\beta).}

If there is an element 1 in \Large{\frak a} such that 1\alpha=\alpha1=\alpha for each \alpha in \Large{\frak a}, we call \Large{\frak a} a linear algebra with identity over F, and call 1 the identity of \Large{\frak a}. The algebra \Large{\frak a} is called commutative if \alpha\beta=\beta\alpha for all \alpha and \beta in \Large{\frak a}.

Definition. Let F[x] be the subspace of F^{\infty} spanned by the vectors 1,x,x^2,\dots. An element of F[x] is called a polynomial over F.

Theorem 1. Let f and g be non-zero polynomials over F. Then
( i ) fg is a non-zero polynomial;
( ii ) \deg (fg)=\deg f+\deg g;
( iii ) fg is a monic polynomial if both f and g are monic polynomials;
( iv ) fg is a scalar polynomial if both f and g are scalar polynomials;
( v ) if f+g\neq 0, \deg (f+g) \leq \max(\deg f,\deg g).
Corollary 1. The set of all polynomials over a given field F equipped with the opertations

\displaystyle{af+bg=(af_0+bg_0,af_1+bg_1,af_2+bg_2,\dots)}

and

\displaystyle{(fg)_n=\sum_{i=0}^nf_jg_{n-i},\qquad n=0,1,2,\dots}

is a commutative linear algebra with identity over F.
Corollary 2. Suppose f,g and h are polynomials over the field F such that f\neq 0 and fg=fh. Then g=h.

Definition. Let \Large{\frak a} be a linear algebra with identity over the field F. We shall denote the identity of \Large{\frak a} by 1 and make the convention that \alpha^0=1 for each \alpha in \Large{\frak a}. Then to each polynomial f=\sum_{i=1}^nf_ix^i over F and \alpha in \Large{\frak a} we associate an element f(\alpha) in \Large{\frak a} by the rule

\displaystyle{f(\alpha)=\sum_{i=1}^nf_i{\alpha}^i}

Theorem 2. Let F be a field and \Large{\frak a} be a linear algebra with identity over F. Supopse f and g are polynomials over F, that \alpha is an element of \Large{\frak a}, and that c belongs to F. Then
( i ) (cf+g)(\alpha)=cf(\alpha)+g(\alpha);
( ii ) (fg)(\alpha)=f(\alpha)g(\alpha).

Lagrange’s interpolation formula: If V=\{f\in F[x]:\deg f\leq n\}+\{0\} and t_0,t_1,\dots,t_n are n+1 distinct elements in F, then for each f\in V, we have

\displaystyle{f=\sum_{i=0}^nf(t_i)P_i,\quad P_i=\prod_{j\neq i}\left(\frac{x-t_j}{t_i-t_j}\right)}

Definition. Let F be a field and let \Large{\mathfrak a} and \Large{\mathfrak a}^{\sim} be linear algebras over F. The algebras \Large{\mathfrak a} and \Large{\mathfrak a}^{\sim} are said to be isomorphic if there is a one-to-one mapping \alpha\to\alpha^{\sim} of \Large{\mathfrak a} onto \Large{\mathfrak a}^{\sim} such that

(c\alpha+d\beta)^{\sim}=c\alpha^{\sim}+d\beta^{\sim} \\ (\alpha\beta)^{\sim}={\alpha}^{\sim}{\beta}^{\sim}

for all \alpha,\beta in \Large{\mathfrak a} and all scalars c,d in F. The mapping \alpha\to\alpha^{\sim} is called an isomorphism of \Large{\mathfrak a} onto \Large{\mathfrak a}^{\sim}. An isomorphism of \Large{\mathfrak a} onto \Large{\mathfrak a}^{\sim} is thus a vector space isomorphism of \Large{\mathfrak a} onto \Large{\mathfrak a}^{\sim} which has the additional property of ‘preserving’ products.

Theorem 3. If F is a field containing an infinite number of distinct elements, the mapping f\to f^{\sim} is an isomorphism of the algebra of polynomials over F onto the algebra of polynomial functions over F.

Lemma. Suppose f and d are non-zero polynomials over a field F such that \deg d\leq \deg f. Then there exists a polynomial g in F[x] such that either f-dg=0 or \deg (f-dg)<\deg f.

Theorem 4. If f,d are polynomials over a field F and d\neq 0 then there exists polynomials q,r\in F[x] such that
( i ) f=dq+r.
( ii ) either r=0 or \deg r<\deg d.
The polynomials q,r satisfying (i) and (ii) are unique.

Definition. Let d be a non-zero polynomial over the field F. If f is in F[x], the preceding theorem shows there is at most one polynomial q in F[x] such that f=dq. If such a q exists we say that d divides f, that f is divisible by d, that f is a multiple of d, and call q the quotient of f and d. We also write q=f/d.
Corollary 1. Let f be a polynomial over the field F, and let c be an element of F. Then f is divisible by x-c if and only if f(c)=0.

Definition. Let F be a field. An element c in F is said to be a root or a zero of a given polynomial f over F if f(c)=0.
Corollary 2. A polynomial f of degree n over a field F has at most n roots in F.

Theorem 5. (Taylor’s Formula) Let F be a field of characteristic zero, c an element of F, and n a positive integer. If f is a polynomial over F and \deg f\leq n, then

\displaystyle{f=\sum_{k=0}^n\dfrac{(D^kf)}{k!}(c)(x-c)^k.}

Theorem 6. Let F be a field of characteristic zero and f a polynomial over F with \deg f\leq n. Then the scalars c is a root of f of multiplicity r if and only if

\displaystyle{(D^kf)(c)=0,\quad 0\leq k\leq r-1\qquad(D^rf)(c)\neq 0}

Definition. Let F be a field. An ideal in F[x] is a subspace M of F[x] such that fg belongs to M whenever f is in F[x] and g is in M. The principle ideal generated by d is M=dF[x].

Theorem 7. If F is a field, and M is any non-zero ideal in F[x], there is a unique monic polynomial d in F[x] such that M is the principle ideal generated by d.
Corollary. If p_1,\dots,p_n are polynomials over a field F, not all of which are 0, there is a unique monic polynomial d in F[x] such that
( a ) d is in the ideal generated by p_1,\dots,p_n;
( b ) d divides each of the polynomials p_i;
Any polynomial satisfying ( a ) and ( b ) necessarily satisfies
( c ) d is divisible by every polynomial which divides each of the polynomials p_1,\dots,p_n.

Definition. If p_1,\dots,p_n are polynomials over a field F, not all of which are 0, the monic generator d of the ideal p_1F[x]+\cdots+p_nF[x] is called the greatest common divisor (g.c.d.) of p_1,\dots,p_n. We say that the polynomials p_1,\dots,p_n are relatively prime if their greatest common divisor is 1, or equivalently if the ideal they generate is all of F[x].

Definition. Let F be a field. A polynomial f in F[x] is said to be reducible over F if there exist polynomials g,h in F[x] of degree \geq 1 such that f=gh, and if not, f is said to be irreducible over F. A non-scalar irreducible polynomial over F is called a prime polynomial over F, and we sometimes say it is a prime in F[x].

Theorem 8. Let p,f and g be polynomials over the field F. Suppose that p is a prime polynomial and that p divides the product fg. Then either p divides f or p divides g.
Corollary. If p is a prime and divides a product f_1,\dots,f_n, then p divides one of the polynomials f_1,\dots,f_n.

Theorem 9. If F is a field, a non-scalar monic polynomial in F[x] can be factored as a product of monic primes in F[x] in one and, except for order, only one way.

Theorem 10. Let f be a non-scalar monic polynomial over the field F and let

\displaystyle{f=p_1^{n_{1}}\cdots p_k^{n_{k}}}

be the prime factorization of f. For each j,1\leq j\leq k, let

\displaystyle{f_j=f/p_j^{n_j}=\prod_{i\neq j}p_i^{n_i}.}

Then f_1,\dots,f_k are relatively prime.

Theorem 11. Let f be a polynomial over the field F with derivative f'. Then f is a product of distinct irreducible polynomials over F if and only if f and f' are relatively prime.

Definition. The field F is called algebraically closed if every prime polynomial over F has degree 1.

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