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- Samuel Otten
- 2007

Let K ≤ F be a normal field extension. Put S := S(K,F) and P := P(K,F). Let K ≤ E ≤ S. (a) We will show that S(E,F) = S. Let a ∈ S. Then a is separable over K. By 5.2.20, a is also separable over E,… (More)

- Samuel Otten
- 2007

Proof. Let F be the field of fractions of R. By 6.2.8, we must show that R = Int(R,F). The inclusion R ⊆ Int(R,F) is immediate. Let f/g ∈ Int(R,F) and let it be written in lowest terms. Suppose to… (More)

- Samuel Otten
- 2006

Certainly 0R ∈ A. Let a, b ∈ A. Then a ∈ An and b ∈ Am for some n,m ∈ N. Without loss of generality, assume n ≤ m. This means An ⊆ Am. So we have a, b ∈ Am. Since Am is a subring, it follows that −a… (More)

- Samuel Otten
- 2006

Hence, we see the existence of π ∈ Sym(I) with α(g) = πg for all g ∈ G. (⇐) Assume π ∈ Sym(I) exists such that α(g)(i) = (π ◦ g ◦ π−1)(i).† By definition of the symmetric group, π : I → I, i → π(i)… (More)

- Samuel Otten
- 2007

Proposition 1. Let R be a ring and M an R-module. Then EndR(M), the set of R-linear maps from M to M , is a subring of End(M). Proof. Recall from 3.1.6 that the ring (End(M), +, ◦) is defined by (α +… (More)

- Zandra de Araujo, Samuel Otten, Salih Birisci
- Educational Technology & Society
- 2017

Flipped instruction is becoming more common in the United States, particularly in mathematics classes. One of the defining characteristics of this increasingly popular instructional format is the… (More)

- Samuel Otten
- 2006

Proof. We consider the two cases: either g ∈ H or g ∈ G−H. Case 1 Suppose g ∈ H. Then for any gh ∈ gH, gh ∈ H. This implies gH ⊆ H. Now, let h be any element of H. Then h = (gg−1)h = g(g−1h), and… (More)

- Samuel Otten
- 2007

(a) Let ¦ : V × S → V, (v, s) → v ¦ s be a function. Define ¦op : Sop × V → V, (s, v) → v ¦ s. Then ¦ is an R-linear right action of S on V if and only if ¦op is an R-linear action of Sop on V . (⇒)… (More)

- Samuel Otten, Beth A. Herbel-Eisenmann
- 2009

This study applied thematic discourse analysis (Lemke, 1990) to a section of a middle school lesson focused on the relationship between the area of parallelograms and rectangles. This analysis… (More)

- Samuel Otten
- 2008

In smooth manifold theory, the notion of a tangent space makes it possible for differentiation to take place on an abstract manifold. In this paper, the notion of a distribution will be presented… (More)