Algebraically in a sentence

A subset S of L is called algebraically independent over K if no non-trivial polynomial relation with coefficients in K exists among the elements of S. The largest cardinality of an algebraically independent set is called the transcendence degree of L/K.

For example, is trivial if k is a finite field or an algebraically closed field (more generally quasi-algebraically closed field ; cf. Tsen's theorem ).

The same argument proves that no subfield of the real field is algebraically closed; in particular, the field of rational numbers is not algebraically closed.

The solution of a pair of linear equations in two variables algebraically – by substitution, by elimination and by cross multiplication.

Algebraically, this involves calculating the discriminant : The curve is non-singular if and only if the discriminant is not equal to zero.

Applying these laws results in a set of simultaneous equations that can be solved either algebraically or numerically.

Either may be used to build general versor operations, but the former has the advantage that it extends to the algebra in a simpler and algebraically more regular fashion.

Every field F has some extension which is algebraically closed.

Extensions Algebraic closure A finite field F is not algebraically closed.

For axiomatization of algebraically closed fields, this is the best possible, as there are counterexamples if a single prime is excluded.

If F is not algebraically closed, let p(x) be a polynomial whose degree is at least 1 without roots.

In fact, it is the smallest algebraically closed field containing the rationals, and is therefore called the algebraic closure of the rationals.

Mohsen Aliabadi generalized Shipman's result for any field in 2013, proving that the sufficient condition for an arbitrary field (of any characteristic) to be algebraically closed is having a root for any polynomial of prime degree.

Not all matrices are diagonalizable (even over an algebraically closed field).

Polynomials of prime degree have roots J. Shipman showed in 2007 that if every polynomial over F of prime degree has a root in F, then every non-constant polynomial has a root in F, thus F is algebraically closed.

Some but not all polynomial equations with rational coefficients have a solution that is an algebraic expression with a finite number of operations involving just those coefficients (that is, it can be solved algebraically ).

The above group can be described algebraically as well as geometrically.

The fields for which the reverse implication holds (that is, the fields such that whenever two polynomials have no common root then they are relatively prime) are precisely the algebraically closed fields.

Third, it is algebraically closed (see above).

Thus, if, say, k is algebraically closed, then all 's are of the form and the above decomposition corresponds to the Jordan canonical form of f. In algebraic geometry, UFD's arise because of smoothness.