Complex number that is a root of a non-zero polynomial in one variable with rational coefficients
The square root of 2 is an algebraic number equal to the length of the hypotenuse of a right triangle with legs of length 1.
An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, , is an algebraic number, because it is a root of the polynomial x2 − x − 1. That is, it is a value for x for which the polynomial evaluates to zero. As another example, the complex number is algebraic because it is a root of x4 + 4.
All integers and rational numbers are algebraic, as are all roots of integers. Real and complex numbers that are not algebraic, such as π and e, are called transcendental numbers.
The set of algebraic numbers is countably infinite and has measure zero in the Lebesgue measure as a subset of the uncountable complex numbers. In that sense, almost all complex numbers are transcendental.
the complex number 1 + i {\displaystyle 1+i} is algebraic because it is a root of x4 + 4. All integers and rational numbers are algebraic, as are all...
their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraicnumber fields and their rings...
of algebraicnumber fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of algebraicnumber theory...
a non-zero algebraicnumber. Then, since eiπ = −1 is algebraic (see Euler's identity), iπ must be transcendental. But since i is algebraic, π must therefore...
the scope of algebra broadened beyond a theory of equations to cover diverse types of algebraic operations and algebraic structures. Algebra is relevant...
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems...
{\displaystyle \mathbb {Q} \left[{\sqrt {-d}}\right]} has class number 1. Equivalently, the ring of algebraic integers of Q [ − d ] {\displaystyle \mathbb {Q} \left[{\sqrt...
(say) is an algebraicnumber. Fields of algebraic numbers are also called algebraicnumber fields, or shortly number fields. Algebraicnumber theory studies...
equations that involve nth roots and, more generally, algebraic expressions. This makes the term algebraic equation ambiguous outside the context of the old...
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are sets with specific operations...
In algebraicnumber theory, an algebraic integer is a complex number that is integral over the integers. That is, an algebraic integer is a complex root...
an important tool and object of study in commutative algebra, algebraicnumber theory and algebraic geometry. The prime ideals of the ring of integers are...
commutative algebra, a major area of modern mathematics. Because these three fields (algebraic geometry, algebraicnumber theory and commutative algebra) are...
In mathematics, an algebraic extension is a field extension L/K such that every element of the larger field L is algebraic over the smaller field K; that...
ideals, and modules over such rings. Both algebraic geometry and algebraicnumber theory build on commutative algebra. Prominent examples of commutative rings...
The roots of such equations are called algebraic numbers – they are a principal object of study in algebraicnumber theory. Compared to Q ¯ {\displaystyle...
algebraic numbers. Consider the approximation of a complex number x by algebraic numbers of degree ≤ n and height ≤ H. Let α be an algebraicnumber of...
similarly. An irrational number may be algebraic, that is a real root of a polynomial with integer coefficients. Those that are not algebraic are transcendental...
In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known...
of an algebraicnumber field is a numerical invariant that, loosely speaking, measures the size of the (ring of integers of the) algebraicnumber field...
\mathbb {Q} } are called algebraicnumber fields, and the algebraic closure of Q {\displaystyle \mathbb {Q} } is the field of algebraic numbers. In mathematical...
branches like algebraicnumber theory and algebraic topology. The word algebra itself has several meanings. Algebraic may also refer to: Algebraic data type...
an algebraic function is a function that can be defined as the root of an irreducible polynomial equation. Algebraic functions are often algebraic expressions...
In abstract algebra, an adelic algebraic group is a semitopological group defined by an algebraic group G over a number field K, and the adele ring A...
abstract development of algebraic geometry. Over finite fields, étale cohomology provides topological invariants associated to algebraic varieties. p-adic Hodge...
having a proper algebraic covering) simple algebraic group defined over a number field is 1. Weil (1959) calculated the Tamagawa number in many cases of...
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