theory, the primitive element theorem states that every finite separable field extension is simple, i.e. generated by a single element. This theorem implies...

(free group), an element of a free generating set Primitive element (Lie algebra), a Borel-weight vector Primitive element theorem Primitive root (disambiguation)...

single element, called a primitive element. Simple extensions are well understood and can be completely classified. The primitive element theorem provides...

restrictive definition of primitive element than that mentioned above after the general normal basis theorem: one requires powers of the element to produce every...

a primitive element of a finite field GF(q) is a generator of the multiplicative group of the field. In other words, α ∈ GF(q) is called a primitive element...

'\mapsto \ell \otimes a\sigma ^{-1}(\ell ').\end{cases}}} The primitive element theorem gives L = K ( α ) {\displaystyle L=K(\alpha )} for some α {\displaystyle...

Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of provability in formal axiomatic theories...

theory from 1928 through 1942, eliminating the dependency on the primitive element theorem. A commutative ring is a set that is equipped with an addition...

0, every finite extension is a simple extension. This is the primitive element theorem, which does not hold true for fields of non-zero characteristic...

0 or r2 = 0. Other ways of determining r1 and r2 are use the primitive element theorem to write K = Q ( α ) {\displaystyle K=\mathbb {Q} (\alpha )} ...

(polynomials) Polynomial remainder theorem (polynomials) Primitive element theorem (field theory) Rational root theorem (algebra, polynomials) Solutions...

Jacobson density theorem is a theorem concerning simple modules over a ring R. The theorem can be applied to show that any primitive ring can be viewed...

In matrix theory, the Perron–Frobenius theorem, proved by Oskar Perron (1907) and Georg Frobenius (1912), asserts that a real square matrix with positive...

order of G, then there exists an element (and thus a cyclic subgroup generated by this element) of order p in G. Theorem (2)—Given a finite group G and...

some element x ∈ K {\displaystyle x\in K} . By the primitive element theorem, there exists such an x {\displaystyle x} , called a primitive element. If...

mathematics, an idempotent element or simply idempotent of a ring is an element a such that a2 = a. That is, the element is idempotent under the ring's...

algebraic number θ ∈ C {\displaystyle \theta \in \mathbb {C} } by the primitive element theorem. α ∈ K is an algebraic integer if there exists a monic polynomial...

strengthened finite Ramsey theorem is then a computable function of n, m, k, but grows extremely fast. In particular it is not primitive recursive, but it is...

denoted GF, then the Stickelberger element of F and the Stickelberger ideal of F can be defined. By the Kronecker–Weber theorem there is an integer m such that...

over Q {\displaystyle \mathbb {Q} } with high probability by the primitive element theorem. If this is the case, we can compute the minimal polynomial q...

single element, called a primitive element, or generating element. The primitive element theorem classifies such extensions. Normal extension An extension...

\{1,\ldots ,n\}} is primitive for every n > 2. Block (permutation group theory) Jordan's theorem (symmetric group) O'Nan–Scott theorem, a classification...

Fermat's little theorem. If a {\displaystyle a} is a primitive element in G F ( q ) {\displaystyle \mathrm {GF} (q)} , then for any non-zero element x {\displaystyle...

integers of K {\displaystyle K} . (This is stronger than the primitive element theorem.) Then, for each integer i ≥ − 1 {\displaystyle i\geq -1} , we...

In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle...

question marks, boxes, or other symbols. In mathematical set theory, Cantor's theorem is a fundamental result which states that, for any set A {\displaystyle...

A246556 in the OEIS) Zsigmondy's theorem Yabuta, Minoru (2001). "A simple proof of Carmichael's theorem on primitive divisors" (PDF). Fibonacci Quarterly...

In mathematical logic, Goodstein's theorem is a statement about the natural numbers, proved by Reuben Goodstein in 1944, which states that every Goodstein...

generated over K by θ (such a θ is guaranteed to exist by the primitive element theorem), and then to examine the minimal polynomial H(X) of θ over K;...

In algebra, a primitive element of a co-algebra C (over an element g) is an element x that satisfies μ ( x ) = x ⊗ g + g ⊗ x {\displaystyle \mu (x)=x\otimes...