The theory of field extensions ( including Galois theory ) involves the roots of polynomials with coefficients in a field ; among other results, this theory leads to impossibility proofs for the classical problems of angle trisection and squaring the circle with a compass and straightedge, as well as a proof of the Abel – Ruffini theorem on the algebraic insolubility of quintic equations.

The term markup is derived from the traditional publishing practice of " marking up " a manuscript, which involves adding handwritten annotations in the form of conventional symbolic printer's instructions in the margins and text of a paper manuscript or printed proof.

The next step in the proof involves a study of the zeros of the zeta function.

The four color theorem states that such colorings are possible for any planar map, but every known proof involves a computational search that is too long to check by hand.

The proof involves a number of deep and original insights which have linked many apparently disparate fields to 3-manifolds.

It is an attractive example of a combinatorial proof ( a proof that involves counting a collection of objects in two different ways ).

) The proof involves Truchet tiles.

Given a commutative diagram, a proof by diagram chasing involves the formal use of the properties of the diagram, such as injective or surjective maps, or exact sequences.

Similarly, Fermat's last theorem is stated in term of elementary arithmetic, which is a part of commutative algebra, but its proof involves deep results of both algebraic number theory and algebraic geometry.

The proof Thomson offered to the latter claim involves what has probably become the most famous example of a supertask since Zeno.

However, the proof involves looking at a Euclidean version of spacetime, in which the time direction is treated as a spatial one, as will be now explained.

" Although this conjecture is true, most of its known proofs depend on the theory of separable and purely inseparable extensions ; for instance, in the case corresponding to the extension being separable, one known proof involves the use of the primitive element theorem in the context of Galois extensions.

* A proof of concept can refer to a partial solution that involves a relatively small number of users acting in business roles to establish whether the system satisfies some aspect of the purpose it was designed for.

A formal petition to de-criminalise acts that temporarily injure a consenting adult was filed with the U. K .' s parliament, then in the Criminal Justice and Immigration Act 2008, S. 66 de-criminalised possession of " pornography " which depicts some acts of injurious sex if it involves oneself ( and potentially others, except for those who cannot or do not consent ), with the burden of proof being on the accused ; Spanner Trust noted their happiness with the consent clause in the Sexual Offences Act 2003.

Close to half of the proof of the Feit – Thompson theorem involves intricate calculations with character values.

In general, if the counting formula involves a division, a similar double counting argument ( if it exists ) gives the most straightforward combinatorial proof of the identity, but double counting arguments are not limited to situations where the formula is of this form.

The simplest proof of given equality involves usage of truth tables:

This involves some sort of interactive proof editor, or other interface, with which a human can guide the search for proofs, the details of which are stored in, and some steps provided by, a computer.

Panel production involves various other chemicals — including wax, dyes, wetting agents, release agents — to make the final product water resistant, fireproof, insect proof, or to give it some other quality.

Pseudoskepticism, by contrast, involves " negative hypotheses "-theoretical assertions that some belief, theory, or claim is factually wrong-without satisfying the burden of proof that such negative theoretical assertions would require.

Pathos is a proof that “ is an appeal to an audience ’ s sense of identity, their self-interest, their emotions ” and because pathos appeals to the deepest parts of the audience ’ s being “ many rhetoricians over the centuries have considered pathos the strongest of the appeals ” because the pathos proof involves “ the power of emotion to sway the mind ” ( Fahnestock 14 ).

In the other direction, the proof is more difficult, and involves showing that in each case ( except the Klein bottle ) a complete graph with a number of vertices equal to the given number of colors can be embedded on the surface.

Indeed, following, suppose ƒ is a complex function defined in an open set Ω ⊂ C. Then, writing for every z ∈ Ω, one can also regard Ω as an open subset of R < sup > 2 </ sup >, and ƒ as a function of two real variables x and y, which maps Ω ⊂ R < sup > 2 </ sup > to C. We consider the Cauchy – Riemann equations at z = 0 assuming ƒ ( z ) = 0, just for notational simplicity – the proof is identical in general case.

The imaginary company had produced a proof of the Riemann Hypothesis but then had great difficulties collecting royalties from mathematicians who had proved results assuming the Riemann Hypothesis.

A simple proof assuming differentiable utility functions and production functions is the following.

Proof by contradiction is also known as indirect proof, apagogical argument, proof by assuming the opposite, and reductio ad impossibilem.

In a proof by contradiction, we start by assuming the opposite, p: that there is a smallest rational number, say, r < sub > 0 </ sub >.

This makes seL4 the first operating-system kernel which closes the gap between trust and trustworthiness, assuming the mathematical proof and the compiler are free from error.

Then, mathematicians started worrying that they were assuming the existence of a continuum of real numbers without proof.

In August 2011, Roger Colbeck and Renato Renner published a proof that any extension of quantum mechanical theory, whether using hidden variables or otherwise, cannot provide a more accurate prediction of outcomes, assuming that observers can freely choose the measurement settings.

These are unproven ; in 1967, Hooley published a conditional proof for the second conjecture, assuming certain cases of the Generalized Riemann hypothesis.

Thus, assuming the Axiom of Choice, we can infer that κ is a strong limit cardinal, which completes the proof of its inaccessibility.

There is then no loss of generality in assuming, since a proof for that case can trivially be adapted for the other case by interchanging and ( leading to the conclusion, which is known to be equivalent to, the desired conclusion.

Instead they turn his argument on its head ( assuming it's valid ) and take it as a proof by contradiction where the possibility of motion is taken for granted.

A proof would have to be a mathematical proof, assuming both the algorithm and specification are given formally.

Grothendieck, with Bombieri, showed the depth of the motivic approach by producing a conditional ( very short and elegant ) proof of the Weil conjectures ( which are proven by different means by Deligne ), assuming the standard conjectures to hold.

Oded Goldreich, et al., took this one step further, showing that, assuming the existence of unbreakable encryption, one can create a zero-knowledge proof system for the NP-complete graph coloring problem with three colors.

would go on to show that, also assuming unbreakable encryption, there are zero-knowledge proofs for all problems in IP = PSPACE, or in other words, anything that can be proved by an interactive proof system can be proved with zero knowledge.

It is a formalization of the common proof technique in which an implication A → B is proved by assuming A and then deriving B from this assumption conjoined with known results.

While these proofs are in some sense " natural ", it can be shown ( assuming a widely believed conjecture on the existence of pseudorandom functions ) that no such proof can possibly be used to solve the P vs. NP problem.

Other related activities include attempts to create lossless data compression algorithms which will compress all possible inputs or to disprove the four-color theorem ; both of these belong to the first category of problems proven to be impossible ( assuming that there is no significant error in the accepted proof of the latter ).

Reductio ad absurdum, reducing to an absurdity, is a method of proof in logic and mathematics, whereby assuming that a proposition is true leads to absurdity ; a proposition is assumed to be true and this is used to deduce a proposition known to be false, therefore the original proposition must have been false.

McCloskey says that most economists when they write are " tendentious ", assuming that they know already, and concentrating on a high-standard of mathematical proof rather than a " scholarly " accumulation of relevant, documented facts about the real world.