Indefinite

The term Indeterminate (Engl. Indeterminate ) is in the math and there especially in abstract algebra for a free generating a polynomial ring or a formal power series ring used. They are listed preferably as capital letters, for example. Or even regardless of a required (unitary) base ring in which the coefficients of the polynomials or power series are the unknowns generate a free monoid ( semigroup with one), which is always written multiplicative and usually commutative needed becomes. ${\displaystyle X,Y}$ ${\displaystyle T.}$ ${\displaystyle R,}$ ${\displaystyle M}$

But even if inverses of elements are added so that there is a (free, commutative or non-commutative) group , one speaks of indeterminate. ${\displaystyle M}$

Viewed in this way, an indeterminate is nothing more than a symbol that (directly or also in its inverse form ) is put together with other such symbols to form sequences of symbols. In the above-mentioned polynomial and power series applications, such a sequence of symbols (a “word” ) “marks” a coefficient from the base ring . Coefficient comparison and calculation rules (such as component-wise addition) refer to this marking. ${\displaystyle X}$ ${\displaystyle X^{-1}}$ ${\displaystyle R}$

An indeterminate can never be the root of a polynomial and in this respect corresponds to a transcendent .

The polynomial ring in the indefinite over is denoted by and the ring of the formal power series is denoted by. ${\displaystyle X}$ ${\displaystyle R}$ ${\displaystyle R[X]}$ ${\displaystyle R[[X]]}$

Monoid, group

For the concatenation of symbols using the (conventional) power notation ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle X}$

${\displaystyle \underbrace {X\cdot \ldots \cdot X} _{n{\text{ Symbole}}}=:X^{n}}$

and has

${\displaystyle \forall m,n\in \mathbb {N} \colon \qquad X^{m}\cdot X^{n}=X^{m+n}.}$

If several indeterminates are involved, then such monomials are in turn linked (written one behind the other). If then the indeterminate commute among each other, one can combine all powers of the same indeterminate into one power in a monomial .

The empty marks

${\displaystyle X^{0}=Y^{0}=T^{0}=1}$

are considered the same. So there is only one empty mark that marks the so-called "constant" link.

If there is a group, i.e. inverses are added, then in the commutative case (as above) one can combine all powers of the same indeterminate to one power. If the concatenation of the indeterminate is to be non-commutative, then the abbreviation rules always apply ${\displaystyle M}$

${\displaystyle \forall u,v\in M\colon \qquad u\cdot X\cdot X^{-1}\cdot v=u\cdot v}$

and

${\displaystyle \forall u,v\in M\colon \qquad u\cdot X^{-1}\cdot X\cdot v=u\cdot v.}$

A polynomial (or a formal power series) corresponds to a mapping

${\displaystyle \alpha \colon M\to R}$

(the "Mark") and is written as ${\displaystyle \sum _{u\in M}\alpha (u)\cdot u.}$

Monomials

These monomials , formed from chained indeterminates, mark a coefficient . For the identity of a polynomial or a formal power series, it is important that all monomials with the same marking are combined (summed up) into a single monomial. ${\displaystyle R}$

In the case of several indeterminates, it can be interesting to consider their role in different variants.

example

The polynomial in two "variables" (the indefinite) ${\displaystyle X,Y}$

${\displaystyle X+}$

${\displaystyle 2\cdot X\cdot Y+}$

${\displaystyle 3\cdot Y}$

has the three monomials above the base ring ${\displaystyle \mathbb {Z} }$

${\displaystyle X,}$

${\displaystyle {\color {Gray}2\cdot }X\cdot Y,}$

${\displaystyle {\color {Gray}3\cdot }Y,}$

the two monomials above the base ring ${\displaystyle \mathbb {Z} [Y]}$

${\displaystyle {\color {Gray}(1+2\cdot Y)\cdot }X,}$

${\displaystyle {\color {Gray}3\cdot Y\cdot }X^{0}}$

and above the base ring the two monomials ${\displaystyle \mathbb {Z} [X]}$

${\displaystyle {\color {Gray}X\cdot }Y^{0},}$

${\displaystyle {\color {Gray}(2\cdot X+3)\cdot }Y}$

each with different coefficients in pale font.

Polynomials

A polynomial in an indeterminate is an expression of the form ${\displaystyle X}$

${\displaystyle a_{0}+a_{1}X+a_{2}X^{2}+\ldots +a_{n}X^{n},}$

where is a non-negative integer and the coefficients are named. A coefficient is “marked” by the (multiplicative) added . The set of all polynomials in (the indeterminate) above a unitary ring is also a (unitary) ring with one, the polynomial ring in above which is denoted by. ${\displaystyle n\in \mathbb {N} _{0}}$ ${\displaystyle a_{i}\in R}$ ${\displaystyle a_{i}:=a_{X^{i}}}$ ${\displaystyle X^{i}}$ ${\displaystyle X}$ ${\displaystyle R}$ ${\displaystyle X}$ ${\displaystyle R,}$ ${\displaystyle R[X]}$

If several, but finitely many, indeterminate factors are involved, then

${\displaystyle R[X_{1},\ldots ,X_{n}]=R[X_{1},\ldots ,X_{n-1}][X_{n}].}$

If it is an infinite set of unknowns, then writing the polynomial as monoid ring with than that generated by the indeterminate Monoid. ${\displaystyle R[M],}$ ${\displaystyle M}$

To express non-commutativity, write and . ${\displaystyle R\langle X,Y\rangle }$ ${\displaystyle R\langle M\rangle }$

Two polynomials are equal if and only if they agree in the coefficients with the same marking.

In contrast, two polynomial functions in an (independent) variable may or may not agree, depending on the value of the variable . ${\displaystyle X}$ ${\displaystyle X}$

The formalism of addition and multiplication of polynomials - and the relationship between polynomial and polynomial function - is used in the

→ Main article : Polynomial ring

described.

Formal power series

A formal power series in an indeterminate is an expression of the form ${\displaystyle X}$

${\displaystyle a_{0}+a_{1}X+a_{2}X^{2}+\ldots ,}$

in which, in contrast to the polynomials, an infinite number of coefficients can differ from 0. The set of all formal power series in (the indeterminate) above a unitary ring is also a (unitary) ring, the ring of formal power series in above which is denoted by. ${\displaystyle a_{i}\in R}$ ${\displaystyle X}$ ${\displaystyle R}$ ${\displaystyle X}$ ${\displaystyle R,}$ ${\displaystyle R[[X]]}$

If several, but finitely many, indeterminate factors are involved, then

${\displaystyle R[[X_{1},\ldots ,X_{n}]]=R[[X_{1},\ldots ,X_{n-1}]][[X_{n}]].}$

In the case of an arbitrarily (possibly infinite) number of indeterminates, the designation is found in the non-commutative case . ${\displaystyle R\langle \langle M\rangle \rangle }$

Two formal power series are equal if and only if they agree in all coefficients.

The formalism of addition and multiplication of formal power series is used in the

→ Main article : Formal power series

explained.

These power series are nicknamed "formal" because, by definition, convergence is not important. If powers with negative exponents also occur, one speaks of formal Laurent series .

For a (ring) object to be able to be "inserted" into a formal power series in , some prerequisites regarding completeness of and convergence of the must be fulfilled. For real and complex , these are im ${\displaystyle x\in R}$ ${\displaystyle X}$ ${\displaystyle R}$ ${\displaystyle a_{i}}$ ${\displaystyle R}$

→ Main article : Power series

described.

Remarks

1. ↑ The quaternions can be constructed as a factor ring of a non-commutative polynomial ring in the three indeterminate modulo the (bilateral) ideal generated by the Hamilton rules (as additional reduction rules) . ${\displaystyle \mathbb {R} \langle I,J,K\rangle }$ ${\displaystyle I,J,K}$
2. ↑ Helmut Koch: Algebraic Number Theory  (= Encycl. Math. Sci.), 2nd printing of 1st. Edition, Volume 62, Springer-Verlag , 1997, ISBN 3-540-63003-1 , p. 167.

literature

• Vollrath, Algebra in the Secondary School, BI Wissenschaftsverlag, Mannheim, Leipzig, Vienna, Zurich 1994, p. 68, ISBN 3-411-17491-9