Algebraic number

Algebraic numbers of the complex plane colored with blue, cyan, red and green according to grade 4, 3, 2 and 1 respectively. The unitary circumference in black.

An algebraic number is any real or complex number that is a solution of an algebraic equation of the form:

anxn+an− − 1xn− − 1+ +a1x+a0=0{displaystyle a_{n}x^{n}+a_{n-1}x^{n-1}+dots +a_{1}x+a_{0}=0,}

Where:

0}" xmlns="http://www.w3.org/1998/Math/MathML">n▪0{displaystyle n/2003/0}0" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27a6a5d982d54202a14f111cb8a49210501b2c96" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;"/>It's the degree of polynomial.

ai한 한 Q{displaystyle a_{i}in mathbb {mathbb {Q} } }, polynomial coefficients are all rational numbers.

anI was. I was. 0{displaystyle a_{n}neq 0}

Examples

• All rational numbers are algebraic because all fraction of the form ab{displaystyle textstyle {frac {a}{b}}}} is solution bx− − a=0{displaystyle bx-a=0}Where a한 한 Z∧ ∧ b한 한 Z{displaystyle ain mathbb {Z} land bin mathbb {Z} }.
• All building numbers are algebraic.
• Some irrational numbers like:2{displaystyle {sqrt {2}}} and 332{displaystyle {frac {sqrt}{3}{3}}{2}}}{2}}} are also algebraic because they are solutions x2− − 2=0{displaystyle x^{2}-2=0} and 8x3− − 3=0{displaystyle 8x^{3}-3=0}respectively.
• Other irrational are not algebraic, as π π {displaystyle pi } (Lindemann, 1882) and e{displaystyle e} (Hermite, 1873). They are, therefore, transcendent.
• The imaginary number i{displaystyle i} is algebraic, being root of x2+1=0{displaystyle x^{2}+1=0,}.

General information

Degree of an algebraic number

It is said that an algebraic number is of degree n{displaystyle n} if it is root of a grade algebraic equation n{displaystyle n}But it's not a grade algebraic equation. n− − 1{displaystyle n-1}.

1− − 3{displaystyle 1-{sqrt {3}}}} is grade two or quadratic irrationality, because it is the root of a second-degree equation, but it is not the root of a first-degree equation.

5− − 3+5{displaystyle 5-{sqrt {3}}+{sqrt {5}}}} is fourth grade (grade 4), as it is the root of a fourth grade equation, but not of a third grade.

Classification

• If a real or complex number is not algebraic, it is said to be transcendent.
• If an algebraic number is a solution to a polynomial degree equation n{displaystyle n}and is not a solution to a lower-grade polynomial equation <math alttext="{displaystyle mm.n{displaystyle m visn}<img alt="{displaystyle mSo it's said to be a grade algebraic number 0)}" xmlns="http://www.w3.org/1998/Math/MathML">n,(n▪0){displaystyle n,(n/20050)}0)}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a3ef5283480cebf8f68c32dbe47dadb04d4c94c" style="vertical-align: -0.838ex; width:9.894ex; height:2.843ex;"/>.

The rational numbers are first grade algebraic numbers, for all rational r=pq;p,q한 한 Z{displaystyle r={frac {p}{q}}}; p,qin mathbb {Z} }, we can always write a grade one polynomial equation with integer coefficients q⋅ ⋅ x− − p=0{displaystyle qcdot x-p=0} whose solution is precisely r{displaystyle r}.

On the other hand, irrational numbers — although they may be algebraic numbers — can never be algebraic grade numbers n=1{displaystyle n=1}.

Properties of the set of algebraic numbers

The set of algebraic numbers is countable, i.e. a bijection can be established with the set of natural numbers.

The sum, difference, product or quotient of two algebraic numbers turns out to be algebraic number, and therefore algebraic numbers constitute an abelian additive group, a ring with unity and a mathematical body. Therefore, the set of algebraic numbers is a subbody of the mathematical body complex numbers. Certainly the sum of a rational number and a radical is an algebraic number; for example 25+73{displaystyle {frac {2}{5}}}+{sqrt[{3}{7}}}}}}}}.

So if s{displaystyle s} and t{displaystyle t} are algebraic numbers. s+t{displaystyle s+t} and s⋅ ⋅ t{displaystyle scdot t}for; s{displaystyle s} There is algebraic number − − s{displaystyle} such as s+(− − s)=0{displaystyle s+(-s)=0}for; sI was. I was. 0{displaystyle sneq 0} There exists s♫{displaystyle s’} such as s⋅ ⋅ s♫=1{displaystyle scdot s'=1}. 0 is additive identity, 1 multipliative identity. The fundamental theorem of algebra ensures that any polynomial equation, with integer coefficients, has a solution in C, has as many roots as indicated the degree, taking into account that some roots can be repeated, it is not said the format of the algebraic number, in fact calculated by numerical analysis procedure.

As a result of the above, all the numbers that can be written from the rationals using only the arithmetic operations sum, difference, product and division whose symbols are +,− − ,× × ,/{displaystyle +,-,times/} respectively, as well as the powers and roots, are algebraic. However, there are algebraic numbers that cannot, in all cases, be written this way, and are all of a greater or equal grade 5. This is a consequence of the Theory of Galois.

It can be shown that if the coefficients ai{displaystyle a_{i}} are any algebraic numbers, the solution of the equation will again be an algebraic number. In other words, the body of algebraic numbers is algebraicly closed. In fact, algebraic numbers are the smallest algebraicly enclosed body containing the rational (its algebraic closure). The set of algebraic numbers, sometimes denoted as A{displaystyle mathbb {A} }, forms a body with the addition and multiplication inherited from the complexes C{displaystyle mathbb {C} }. Unlike complex numbers algebraic numbers are a numberable set and therefore its cardinal is alef 0. This is a consequence of the number of polynomials with integer coefficients.

Algebraic integers

An algebraic number that satisfies a polynomial degree equation n{displaystyle n} with an=1{displaystyle a_{n}=1} is called algebraic integer. Some examples of algebraic integers are: 3⋅ ⋅ 212+5{displaystyle 3cdot 2^{frac {1}{2}} +5}, 6i− − 2{displaystyle 6i-2}. The sum, difference and product of algebraic integers becomes an algebraic integer, which means algebraic integers form a ring. The name of algebraic integer comes from the fact that the only rational numbers that are algebraic integers are the integers themselves.

Algebraic Extensions

The notions of algebraic number and algebraic integer can be generalized to other fields, not only to that of complexes; see algebraic extension.

In general, if we have two bodies (K,+,⋅ ⋅ ){displaystyle (K,+,cdot)} and (L,+,⋅ ⋅ ){displaystyle (L,+,cdot)} so that the second is extension of the first, we will say that α α 한 한 L{displaystyle alpha in L} is algebraic about K{displaystyle K} if there is a polynomial p한 한 K[chuckles]x]{displaystyle pin K[x]} of the α α {displaystyle alpha ,} is root (p(α α )=0{displaystyle p(alpha)=0,}).

History

Leonhard Euler divided numbers into algebraic and transcendental numbers in 1748. In 1844 Liouville obtained the first criterion necessary for a number to be algebraic, and therefore a sufficient criterion to be a transcendental number. The general theory of integer algebraic numbers was made, almost at the same time, by Dedekind (1877 -1895) and Zolotariov (1874). The foundation of this theory was built by Kummer.