scholarly journals On Monomorphisms and Subfields

2019 ◽  
Vol 27 (2) ◽  
pp. 133-137
Author(s):  
Christoph Schwarzweller
Keyword(s):  
Irreducible Polynomial ◽  
Field Extension ◽  
Isomorphic Copy ◽  
Field Extensions ◽  
The Third ◽  
Fourth Part ◽  
Roots Of Polynomials ◽  
Classical Proof ◽  
Do So

Summary This is the second part of a four-article series containing a Mizar [2], [1] formalization of Kronecker’s construction about roots of polynomials in field extensions, i.e. that for every field F and every polynomial p ∈ F [X]\F there exists a field extension E of F such that p has a root over E. The formalization follows Kronecker’s classical proof using F [X]/<p> as the desired field extension E [5], [3], [4]. In the first part we show that an irreducible polynomial p ∈ F [X]\F has a root over F [X]/<p>. Note, however, that this statement cannot be true in a rigid formal sense: We do not have F ⊆ [X]/ < p > as sets, so F is not a subfield of F [X]/<p>, and hence formally p is not even a polynomial over F [X]/ < p >. Consequently, we translate p along the canonical monomorphism ϕ : F → F [X]/<p> and show that the translated polynomial ϕ (p) has a root over F [X]/<p>. Because F is not a subfield of F [X]/<p> we construct in this second part the field (E \ ϕF )∪F for a given monomorphism ϕ : F → E and show that this field both is isomorphic to F and includes F as a subfield. In the literature this part of the proof usually consists of saying that “one can identify F with its image ϕF in F [X]/<p> and therefore consider F as a subfield of F [X]/<p>”. Interestingly, to do so we need to assume that F ∩ E = ∅, in particular Kronecker’s construction can be formalized for fields F with F ∩ F [X] = ∅. Surprisingly, as we show in the third part, this condition is not automatically true for arbitray fields F : With the exception of 𝕑2 we construct for every field F an isomorphic copy F′ of F with F′ ∩ F′ [X] ≠ ∅. We also prove that for Mizar’s representations of 𝕑n, 𝕈 and 𝕉 we have 𝕑n ∩ 𝕑n[X] = ∅, 𝕈 ∩ 𝕈 [X] = ∅ and 𝕉 ∩ 𝕉 [X] = ∅, respectively. In the fourth part we finally define field extensions: E is a field extension of F iff F is a subfield of E. Note, that in this case we have F ⊆ E as sets, and thus a polynomial p over F is also a polynomial over E. We then apply the construction of the second part to F [X]/<p> with the canonical monomorphism ϕ : F → F [X]/<p>. Together with the first part this gives - for fields F with F ∩ F [X] = ∅ - a field extension E of F in which p ∈ F [X]\F has a root.

scholarly journals On Roots of Polynomials over F[X]/ 〈p〉

2019 ◽  
Vol 27 (2) ◽  
pp. 93-100
Author(s):  
Christoph Schwarzweller
Keyword(s):  
Irreducible Polynomial ◽  
Field Extension ◽  
Isomorphic Copy ◽  
Field Extensions ◽  
The Third ◽  
Fourth Part ◽  
Roots Of Polynomials ◽  
Classical Proof ◽  
Do So

Summary This is the first part of a four-article series containing a Mizar [3], [1], [2] formalization of Kronecker’s construction about roots of polynomials in field extensions, i.e. that for every field F and every polynomial p ∈ F [X]\F there exists a field extension E of F such that p has a root over E. The formalization follows Kronecker’s classical proof using F [X]/<p> as the desired field extension E [9], [4], [6]. In this first part we show that an irreducible polynomial p ∈ F [X]\F has a root over F [X]/<p>. Note, however, that this statement cannot be true in a rigid formal sense: We do not have F ⊆ [X]/ < p > as sets, so F is not a subfield of F [X]/<p>, and hence formally p is not even a polynomial over F [X]/ < p >. Consequently, we translate p along the canonical monomorphism ϕ: F → F [X]/<p> and show that the translated polynomial ϕ(p) has a root over F [X]/<p>. Because F is not a subfield of F [X]/<p> we construct in the second part the field (E \ ϕF )∪F for a given monomorphism ϕ : F → E and show that this field both is isomorphic to F and includes F as a subfield. In the literature this part of the proof usually consists of saying that “one can identify F with its image ϕF in F [X]/<p> and therefore consider F as a subfield of F [X]/<p>”. Interestingly, to do so we need to assume that F ∩ E =∅, in particular Kronecker’s construction can be formalized for fields F with F \ F [X] =∅. Surprisingly, as we show in the third part, this condition is not automatically true for arbitray fields F : With the exception of 𝕑2 we construct for every field F an isomorphic copy F′ of F with F′ ∩ F′ [X] ≠∅. We also prove that for Mizar’s representations of 𝕑n, 𝕈 and 𝕉 we have 𝕑n ∩ 𝕑n[X] = ∅, 𝕈 ∩ 𝕈[X] = ∅and 𝕉 ∩ 𝕉[X] = ∅, respectively. In the fourth part we finally define field extensions: E is a field extension of F i F is a subfield of E. Note, that in this case we have F ⊆ E as sets, and thus a polynomial p over F is also a polynomial over E. We then apply the construction of the second part to F [X]/<p> with the canonical monomorphism ϕ : F → F [X]/<p>. Together with the first part this gives - for fields F with F ∩ F [X] = ∅ - a field extension E of F in which p ∈ F [X]\F has a root.

scholarly journals On the Intersection of Fields F with F [X]

2019 ◽  
Vol 27 (3) ◽  
pp. 223-228
Author(s):  
Christoph Schwarzweller
Keyword(s):  
Irreducible Polynomial ◽  
Field Extension ◽  
Isomorphic Copy ◽  
Field Extensions ◽  
The Third ◽  
Fourth Part ◽  
Roots Of Polynomials ◽  
Classical Proof ◽  
Do So

Summary This is the third part of a four-article series containing a Mizar [3], [1], [2] formalization of Kronecker’s construction about roots of polynomials in field extensions, i.e. that for every field F and every polynomial p ∈ F [X]\F there exists a field extension E of F such that p has a root over E. The formalization follows Kronecker’s classical proof using F [X]/<p> as the desired field extension E [6], [4], [5]. In the first part we show that an irreducible polynomial p ∈ F [X]\F has a root over F [X]/<p>. Note, however, that this statement cannot be true in a rigid formal sense: We do not have F ⊆ F [X]/ < p > as sets, so F is not a subfield of F [X]/<p>, and hence formally p is not even a polynomial over F [X]/ < p >. Consequently, we translate p along the canonical monomorphism ϕ: F → F [X]/<p> and show that the translated polynomial ϕ (p) has a root over F [X]/<p>. Because F is not a subfield of F [X]/<p> we construct in the second part the field (E \ ϕF)∪F for a given monomorphism ϕ: F → E and show that this field both is isomorphic to F and includes F as a subfield. In the literature this part of the proof usually consists of saying that “one can identify F with its image ϕF in F [X]/<p> and therefore consider F as a subfield of F [X]/<p>”. Interestingly, to do so we need to assume that F ∩ E = ∅, in particular Kronecker’s construction can be formalized for fields F with F ∩ F [X] = ∅. Surprisingly, as we show in this third part, this condition is not automatically true for arbitrary fields F : With the exception of ℤ2 we construct for every field F an isomorphic copy F′ of F with F′ ∩ F′ [X] ≠ ∅. We also prove that for Mizar’s representations of ℤn, ℚ and ℝ we have ℤn ∩ ℤn[X] = ∅, ℚ ∩ ℚ[X] = ∅ and ℝ ∩ ℝ[X] = ∅, respectively. In the fourth part we finally define field extensions: E is a field extension of F iff F is a subfield of E. Note, that in this case we have F ⊆ E as sets, and thus a polynomial p over F is also a polynomial over E. We then apply the construction of the second part to F [X]/<p> with the canonical monomorphism ϕ: F → F [X]/<p>. Together with the first part this gives – for fields F with F ∩ F [X] = ∅ – a field extension E of F in which p ∈ F [X]\F has a root.

scholarly journals Field Extensions and Kronecker’s Construction

2019 ◽  
Vol 27 (3) ◽  
pp. 229-235
Author(s):  
Christoph Schwarzweller
Keyword(s):  
Irreducible Polynomial ◽  
Field Extension ◽  
Isomorphic Copy ◽  
Field Extensions ◽  
The Third ◽  
Fourth Part ◽  
Roots Of Polynomials ◽  
Classical Proof ◽  
Do So

Summary This is the fourth part of a four-article series containing a Mizar [3], [2], [1] formalization of Kronecker’s construction about roots of polynomials in field extensions, i.e. that for every field F and every polynomial p ∈ F [X]\F there exists a field extension E of F such that p has a root over E. The formalization follows Kronecker’s classical proof using F [X]/<p> as the desired field extension E [6], [4], [5]. In the first part we show that an irreducible polynomial p ∈ F [X]\F has a root over F [X]/<p>. Note, however, that this statement cannot be true in a rigid formal sense: We do not have F ⊆ F [X]/ < p > as sets, so F is not a subfield of F [X]/<p>, and hence formally p is not even a polynomial over F [X]/ < p >. Consequently, we translate p along the canonical monomorphism ϕ: F → F [X]/<p> and show that the translated polynomial ϕ (p) has a root over F [X]/<p>. Because F is not a subfield of F [X]/<p> we construct in the second part the field (E \ ϕF)∪F for a given monomorphism ϕ: F → E and show that this field both is isomorphic to F and includes F as a subfield. In the literature this part of the proof usually consists of saying that “one can identify F with its image ϕF in F [X]/<p> and therefore consider F as a subfield of F [X]/<p>”. Interestingly, to do so we need to assume that F ∩ E = ∅, in particular Kronecker’s construction can be formalized for fields F with F ∩ F [X] = ∅. Surprisingly, as we show in the third part, this condition is not automatically true for arbitrary fields F : With the exception of ℤ2 we construct for every field F an isomorphic copy F′ of F with F′ ∩ F′ [X] ≠ ∅. We also prove that for Mizar’s representations of ℤn, ℚ and ℝ we have ℤn ∩ ℤn[X] = ∅, ℚ ∩ ℚ[X] = ∅ and ℝ ∩ ℝ[X] = ∅, respectively. In this fourth part we finally define field extensions: E is a field extension of F iff F is a subfield of E. Note, that in this case we have F ⊆ E as sets, and thus a polynomial p over F is also a polynomial over E. We then apply the construction of the second part to F [X]/<p> with the canonical monomorphism ϕ: F → F [X]/<p>. Together with the first part this gives – for fields F with F ∩ F [X] = ∅ – a field extension E of F in which p ∈ F [X]\F has a root.

Sarakhsi’s Doctrine of Juristic Preference (Istihsan) as a Methodological Approach Toward Worldly Affairs (Ahkam al-Dunya)

1988 ◽  
Vol 5 (2) ◽  
pp. 181-203
Author(s):  
Husain Kassim
Keyword(s):  
Methodological Approach ◽  
The Third ◽  
Fourth Part

In the present investigation, we shall develop systematically Sarakhsrsdoctrine of Juristic preference from his Mabsut, Usul and Bab al-Muwada'aof Sharh al-Siyar al Kabir and demonstrate how Sarakhsi establishes itsrelevance as a methodological approach toward worldly affairs.The investigation is carried out in four parts:In the first part, we shall relate Sarakhsi’s doctrine of juristic preference(istihan) with his concept of treaties (muwada'a). According to Sarakhsimuwada'a is an autonomous discipline and its main focus is worldly affairsas relations (muamalat) of Muslims with other nations.In the second part, it is investigated how Sarakhsi strives to see thejustification for the application of the doctrine of juristic preference to itindependently of the doctrine of systematic reasoning (qiyas) by establishingthe ’illa (effective reasoning) of the doctrine of juristic preference on the basisof asl derived from the Qur’an and Hadith.In the third part, we shall discuss how Sarakhsi systematizes the doctrineof juristic preference by analyzing the ’illa employed by it in various formsand shows that it is connected with asl.Finally, in the fourth part, we shall show how Sarakhsi justifies theemployment of the doctrine of juristic preference as a methodological approachtoward muwadah and worldly affairs ...

scholarly journals A political ontology for Europe: Roberto Esposito’s instituent paradigm

2021 ◽  
Author(s):  
Rita Fulco
Keyword(s):  
Political Theology ◽  
Political Crisis ◽  
Recent Proposal ◽  
Political Ontology ◽  
The Third ◽  
European Crisis ◽  
Do So

AbstractThe aim of my article is to relate Roberto Esposito’s reflections on Europe to his more recent proposal of instituent thought. I will try to do so by focusing on three theoretical cornerstones of Esposito’s thought: the first concerns the evidence of a link between Europe, philosophy and politics. The second is deconstructive: it highlights the inadequacy of the answers of the most important contemporary ontological-political paradigms to the European crisis, as well as the impossibility of interpreting this crisis through theoretical-political categories such as sovereignty. The third relates more directly to the proposal of a new political ontology, which Esposito defines as instituent thought. Esposito’s discussion of political theology is the central theoretical nucleus of this study. This discussion will focus, in particular, on the category of negation, from which any political ontology that is based on pure affirmativeness or absolute negation is criticized. In his opinion, philosophical theories developed on the basis of these assumptions have proved to be incomplete or ineffective in relation to the current European and global philosophical and political crisis. Esposito therefore perceives the urgent need to propose a line of thought that is neither negatively destituent (post-Heideggerian), nor affirmatively constituent (post-Deleuzian, post-Spinozian), but instituent (neo-Machiavellian), capable of thinking about order through conflict (the affirmative through the negative). Provided that we do not think of the institution statically–in a conservative sense–but dynamically, as constant instituting in which conflict can become an instrument of a politics increasingly inspired by justice.

scholarly journals Art Criticism and the State of Feminist Art Criticism

Arts ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 28
Author(s):  
Katy Deepwell
Keyword(s):  
Women Artists ◽  
Art Criticism ◽  
The State ◽  
Art World ◽  
Feminist Art ◽  
The Gaze ◽  
The Third ◽  
Fourth Part ◽  
State Of Art ◽  
First Offers

This essay is in four parts. The first offers a critique of James Elkins and Michael Newman’s book The State of Art Criticism (Routledge, 2008) for what it tells us about art criticism in academia and journalism and feminism; the second considers how a gendered analysis measures the “state” of art and art criticism as a feminist intervention; and the third, how neo-liberal mis-readings of Linda Nochlin and Laura Mulvey in the art world represent feminism in ideas about “greatness” and the “gaze”, whilst avoiding feminist arguments about women artists or their work, particularly on “motherhood”. In the fourth part, against the limits of the first three, the state of feminist art criticism across the last fifty years is reconsidered by highlighting the plurality of feminisms in transnational, transgenerational and progressive alliances.

Ancient Hinduism as a Missionary Religion

Numen ◽  
1992 ◽  
Vol 39 (2) ◽  
pp. 175-192 ◽  
Author(s):  
Arvind Sharma
Keyword(s):  
Tenth Century ◽  
Fifth Century ◽  
Historical Material ◽  
Textual Evidence ◽  
The Third ◽  
Fourth Part ◽  
History Of

AbstractThe paper is conceptually divided into four parts. In the first part the widely held view that ancient Hinduism was not a missionary religion is presented. (The term ancient is employed to characterize the period in the history of Hinduism extending from fifth century B.C.E. to the tenth century. The term 'missionary religion' is used to designate a religion which places its followers under an obligation to missionize.) In the second part the conception of conversion in the context of ancient Hinduism is clarified and it is explained how this conception differs from the notion of conversion as found in Christianity. In the third part the view that ancient Hinduism was not a missionary religion is challenged by presenting textual evidence that ancient Hinduism was in fact a missionary religion, inasmuch as it placed a well-defined segment of its members under an obligation to undertake missionary activity. Such historical material as serves to confirm the textual evidence is then presented in the fourth part.

A European Perspective on Religion and Welfare: Contrasts and Commonalities

2012 ◽  
Vol 11 (4) ◽  
pp. 589-599 ◽  
Author(s):  
Grace Davie
Keyword(s):  
Social Policy ◽  
The State ◽  
Comparative Perspective ◽  
European Perspective ◽  
The Third ◽  
Effective Delivery ◽  
Delivery Of Services ◽  
Do So ◽  
West European

This article places the British material on religion and social policy in a comparative perspective. In order to do so, it introduces a recently completed project on welfare and religion in eight European societies, entitled ‘Welfare and Religion in a European Perspective’. Theoretically it draws on the work of two key thinkers: Gøsta Esping-Andersen and David Martin. The third section elaborates the argument: all West European societies are faced with the same dilemmas regarding the provision of welfare and all of them are considering alternatives to the state for the effective delivery of services. These alternatives include the churches.

The Irish–British Dimension

2021 ◽  
pp. 106-126
Author(s):  
Paul Gillespie
Keyword(s):  
Northern Ireland ◽  
Interstate Relations ◽  
The Third ◽  
Ability To Act ◽  
Political Relations ◽  
Power Scale ◽  
Opening Up ◽  
Normal Relationship ◽  
Do So

Power, scale, and wealth have moulded relations between Ireland and Britain historically and will continue to do so in future. Political relations between them have been determined by these asymmetric factors, giving much greater strength to the larger and richer island. Nevertheless, both islands exist within a larger European and transatlantic setting, a geopolitical fact that can mitigate or counteract Britain’s ability to act exclusively in its own interests. The chapter first explores this history and structure of the Irish–British relationship and then examines current political relations between the two islands, as seen in the intense joint efforts to bring peace to Northern Ireland and to regularize their interstate relations. Brexit rudely interrupts that new more normal relationship, as the third section argues, opening up several scenarios for changing constitutional futures within and between the two islands explored in the final one.

The Gathering of A Community: The British-born of San Francisco in 1852 and 1872

1976 ◽  
Vol 10 (3) ◽  
pp. 279-312 ◽  
Author(s):  
R. A. Burchell
Keyword(s):  
Nineteenth Century ◽  
San Francisco ◽  
High Rate ◽  
Geographical Mobility ◽  
The Third ◽  
Human Effort ◽  
The Subject ◽  
Incomplete Picture ◽  
Do So

Studies of the Massachusetts communities of Newburyport and Boston have revealed a high rate of geographical mobility for their populations, in excess of what had been previously thought. Because of the difficulty in tracing out-migrants these works have concentrated on persisters, though to do so is to give an incomplete picture of communal progress. Peter R. Knights in his study of Boston between 1830 and 1860 attempted to follow his out-migrants but was only able to trace some 27 per cent of them. The problem of out-migration is generally regarded as being too large for solution through human effort, but important enough now to engage the computer. What follows bears on the subject of out-migration, for it is an analysis of where part of the migrating populations of the east went in the third quarter of the nineteenth century, namely to San Francisco.

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