primitive element

Noun

of a finite field of a lattice of a coalgebra over an element g of a free group

(algebra, field theory) An element that generates a simple extension.

Japanese Meaning

単純拡大（体論における）の生成元、すなわち拡大体全体を一つの元によって生成できる要素

What is this buttons?

In the proof, we exhibited a primitive element whose minimal polynomial has degree equal to the extension degree.

Google Translation / DeepL Translation

証明では、単拡大を生成する元（原始元）を構成し、その最小多項式の次数が拡大の次数と等しいことを示した。

Details

Sense（1）

(algebra, field theory) An element that generates a simple extension.

An algebraic extension L/K is called simple if L=K(𝛼) for some 𝛼∈L. The element 𝛼 is called a primitive element for L/K. Every finite separable algebraic field extension is simple.

Suppose that L=K(𝛼₁,…𝛼ᵣ) is a finite separable extension and K_0⊆K is an infinite subset of K. Then there exists a primitive element 𝛼 of the form 𝛼=∑ᵢ₌₁ʳc_i𝛼ᵢ with c_i∈K_0.

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Sense（2）

Furthermore, if the irreducible polynomial has a primitive element 𝛼 (where 𝛼=1) that is a root, then the polynomial is termed a primitive polynomial and corresponds to the polynomial for a maximal length feedback shift register.

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Sense（3）

(number theory) Given a modulus n, a number g such that every number coprime to n is congruent (modulo n) to some power of g; equivalently, a generator of the multiplicative field of integers modulo n.

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Sense（4）

But suppose L'∈C_𝜈(S_0) so that operatorname det(L')=𝜂'𝜋 blacktriangleright 0 for some totally positive unit 𝜂' and so that L' is everywhere locally a primitive element of the o-lattice R_𝜈.

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Sense（5）

(algebra, of a coalgebra over an element g) An element x ∈ C such that μ(x) = x ⊗ g + g ⊗ x, where μ is the comultiplication and g is an element that maps to the multiplicative identity 1 of the base field under the counit (in particular, if C is a bialgebra, g = 1).

A primitive element of a Hopf algebra is an element h∈H such that

𝛥h=1⊗h+h⊗1.

It is easily seen that the bracket [x,y]:=xy-yx of two primitive elements is again a primitive element. It follows that primitive elements form a Lie algebra. For H=U(g) any element of g is primitive and in fact using the Poincaré-Birkhoff-Win theorem, one can show that the set of primitive elements of U(g) coincides with the Lie algebra g.

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